The Gemara in Rosh HaShanah 17b states that HaShem told Moshe Rabbeinu that whenever the Jews sin, they should recite the 13 Middos and they will be forgiven.
Why do we say the 13 Middos so often during the Selichos before Rosh Hashanah, during the days between Rosh Hashanah and Yom Kippur, and on Yom Kippur itself? If HaShem promised us that we just need to say it and we’ll be forgiven, we should just say it once and we’ll be good to go.
See this article by Rabbi Frand, where he discusses the related question of why it is that we do not always see the recital of the 13 middot actually being effective in practice.
He quotes two basic approaches:
The Reishis Chochmah explains that the gemara referred to in the question is not actually talking about reciting the 13 Middos, but emulating them. This is inferred from the phrase ya-asu lefanai k’seder hazeh literally that they should do (i.e. act like) this list of middot.
The Imrei Binah mantains that the gemara is indeed referring to reciting the middot, but that is only a necessary condition. In addition, one needs to have the mindset of a "shaliach tzibbur" i.e. to be praying on behalf of others as well as oneself.
I think you can use either of these approaches to answer the question here:
We repeat the 13 middot over and over during Elul and the Yamim Noraim to drill into ourselves these attributes of Hashem which we are to emulate.
We repeat them to give ourselves multiple chances of reciting them (at least once) with the feelings and prayers for others, which are required for them to "work".
When we say the 13 Middos, we are promised that they will not go unanswered, not that we will be forgiven for all of our sins. Even if our sins are not forgiven, they will still help in some other way (like healing a sick person, etc.) So we say the 13 Middos as often as we can so that maybe some of them will forgive our sins.
