As msh210 already pointed out, the form שִׁלְחָה is not the feminine construct, it is actually the masculine emphatic imperative. As to why Tanach uses the emphatic imperative rather than the normal one, there is an excellent article by Fassberg which discusses this. It appears that the emphatic imperative שִׁלְחָה is used when the action of the verb is directed towards the speaker. On the other hand, when the action of verb is not directed at the speaker, the regular imperative שְׁלַח tends to be used. The example you gave of II Kings 4:22 perfectly demonstrates this:
וַתִּקְרָא אֶל אִישָׁהּ וַתֹּאמֶר שִׁלְחָה נָא לִי ...
Here the action of the verb is coming back to the speaker. We can also see that the widow chose to include the word נָא which indicates further emphasis on doing the action.
There are other examples in the Semitic languages where adding a letter to the end of a word denotes emphasis. If you can recall the Aramaic sections of Tanach, you might remember that nouns have an emphatic form in addition to their normal form. So מֶּלֶךְ means "a king" but מַלְכָּא means "the king." The aleph as the end of the word is a marker for emphasis.