In 1980 I published an article in the Baltimore Jewish Times called, "What the Matchmaker Did and Didn't Do for Me." I interviewed many matchmakers and their clients and made myself (then single, of course) a guinea pig for a Boro Park matchmaker. The matchmakers make it pretty clear, that they take the time to find someone who is appropriate for you, and then, if there is an engagement, he/she gets paid on the engagement. Back then the fee was $1,000 to $5,000. I have no idea what it is now. My matchmaker also stipulated that I could only go on three dates -- if we don't feel the spark after three dates, then it's over -- move on to someone else -- "I'm not in the business of making you friends," she said. It was clear also that if I make an engagement that doesn't make it to the chuppah, too bad -- no refunds; she did her job, and may have even had expenses (some matchmakers work internationally, finding matches between couples in different countries). Since then I've become an attorney.
A bit on the law: In both American common law and Jewish halacha, a contract consists of an offer and acceptance, a meeting of the minds, and consideration of some sort before a contract can be deemed to be effective, although there are exceptions to the rule. In Jewish law and American law, "consideration" is made by such things as a down payment, acceptance of loan money, or even the acceptance of a key to an apartment you just agreed to let. The same is true in Jewish law, where also a kinyan can be effected through formalities such as the performance of an act representing the receipt of the subject matter of the transaction by the purchaser, donee, hirer, or borrower; See Baba Metzia 99a.) An example of the latter may be that when you complete the "sale" of your chametz through the rabbi, the ceremonial holding by both parties of a pen or handkerchief, for example, completes the kinyan.
Let's assume that these same conditions I received were given to you, and lets assume that unlike my matchmaker, you actually had a written contract (this makes it easier for analysis in American and Jewish law), and paid a $100 consideration for expenses. Since you have consideration, you have an effective kinyan and a contract is made. So if there is no dispute about the terms, and the matchmaker introduced you to your bride, as required, then payment is due on the announcement of the engagement.
There is also a body of Jewish law that applies customary practices to the laws of payment to a shadchan.
Like any worker who provides customers with a service, a shadchan must be paid for his work, which consists of arranging shidduchim. A shadchan is similar to the various types of brokers who earn fees for bringing together two sides in a transaction. See Rama, Choshen Mishpat 87:39 and 185:10.
A shadchan whose work is solicited by the family is like any hired worker, whose payment is in accords with whatever contract or agreement they parties had. See Shulchan Aruch, Choshen Mishpat 332 and elsewhere. One who volunteers to be a shadchan for a particular person, or makes an unsolicited suggestion for a shiddach, at best is entitled only to the value of the service, which may vary. See Ibid and Ponim Me’iros vol II #63 [regarding a shidduch in which the chosson and kallah lived in different cities, which had different customs regarding how much a shadchan was paid], cited in Pischei Teshuvah ibid.; Pischei Choshen, Sechirus 14:1 [see there for a number of additional sources]. Thus, while a shadchan must be paid whether or not his services were solicited,4 the nature of his involvement impacts several common questions, as will be discussed below.
Whether the shadchan’s fee is due upon the couple’s engagement or only after their marriage is dependent on the terms of any agreement between the shadchan and the family, or the prevalent custom in the place that his service is provided. Where no custom exists, the payment may be delayed until after the wedding. It appears that the custom today in most places (if not all) is to pay the shadchan after the engagement. Rama, Choshen Mishpat 185:10.
Despite your circumstances, the Shadchen did what he was contracted to do. Like any hired worker, the Torah requires that shadchan be paid for his work on time. See Vayikra 19:13 and Devorim 24:15; Shulchan Aruch, Choshen Mishpat 339.17. If the shadchan merely volunteered his services, then, depending upon the custom of the community, he or she should be paid for the value of the service pursuant to local custom -- not his actual work. Since he is not technically a worker, in that case, there is no specific time payment is required. Rav Yosef Shalom Elyashiv zt”l ibid.; Halichos Yisroel #2; Lehoros Nosson ibid.; see Pischei Choshen ibid.
Where the custom is that a shadchan is paid on engagement, then whether or not the wedding is later called off, he is still entitled to payment. Taz, Choshen Mishpat 185:26; Bais Shmuel, Even Ha’ezer 50:23; see Rama ibid. As my shadchan told me, she "wasn't in the business" of making me friends; if after three dates, I didn't see marriage as a sure thing, I should move on. However, I looked into it and found that in some communities, the custom is to pay the shadchan at the time of chuppah. I wonder what rights the shadchan has if the couple decide to go "modern" and just live together until they're both sure they want to be married? (Yes, I know, it would be a scandal.) For other issues involving payment for services by shadchanim, see "Paying the Shadchan: Who, When, and How Much?", Yeted Ne'eman, Dec. 24, 2014.
Under American law, even if there was no written contract, going to a person who is in the business of arranging marriages, and asking him or her to look for someone is a commitment to a contract. If the shadchan finds someone appropriate, you are obligated to pay for the service. The classic law school case of an implied contract such as this, is sitting in a barber's chair; you can't expect the barber not to cut your hair then for free even though you did not negotiate a contract to do so. The fundamental here is simple. Normally, contract law requires formal offer and acceptance and a meeting of the minds. But if it can be shown that a party voluntarily accepts and avails himself of valuable services rendered for his benefit, when he has the option whether to accept or reject them, even if there is no distinct proof that they were rendered by his authority or request, a promise to pay for them may be inferred. Abbot v. Hermon, 7 Me 118 (1830), cited in Fuller and Eisenberg, Basic Contract Law, 4th Ed., p. 472 (West 1981).
As far as your case: Did anyone in your family contract with the shadchan? If so, is there any doubt that the shadchan found you a wife? Was there any deadline or time limit agreed to in the contract? If the answer to each of these questions is "no" it appears you are liable to the shadchan for the fee that was agreed upon. The shadchan has a business and devoted part of his business hours looking for your bride. Yet he was incentivized to look out for you because your family's promise of payment upon engagement or chuppah. He took the risk that he'd guess wrongly about the girl for you. You took the risk that if he found your bride you owed him money. Furthermore, if the money was really an issue, you could have said no to the engagement, wish the girl well and move on. So there wasn't much risk for you, as long as the shadchan didn't find the girl meant just for you. If he did, no matter what you will pay, it was a bargain.
So, in closing, mazel tov on your engagement. My blessing to you is that you should have a very good paranasah (earnings from a job or business), enough so that you can pay yeshiva tuition for many children, pay off the shadchan, and build a bayit nee'man.