Can I pronounce 十三日 as じゅうみっか?
Can you? Yes. You could also pronounce it as wagawaga boom tash, or as fweeeEEEEEE gobble gobble. However, none of these are likely to be understood by Japanese speakers to mean "the thirteenth day of the month".
Windows IME seems to accept it
The Windows IME is probably parsing this as じゅう + みっか, not as じゅうみっか as a single term.
In fact, typing this here just now, I see that the IME has zero kanji conversion candidates for the whole kana string じゅうみっか -- only for the pieces of it, with じゅう as one sub-string and みっか as the other. If I instead enter the kana string じゅうさんにち, I do indeed see 十三日 as a conversion candidate for the entire kana string, not just a part of it.
十四日 is pronounced as じゅうよっか, matching the pronunciation of 四日, which is よっか
一日 is another irregular reading for dates, ついたち and never いちにち. Other dates up through "the tenth" use the native Japanese-derived readings. 二十日 for dates is also native-derived, and is read as はつか, never にじゅうにち.
All of the "fours" in dates are strange for some reason, related to obscure details of historical development. This holds true for 四日 (only よっか, never よんにち or しにち), 十四日 (only じゅうよっか, never じゅうよんいち or じゅうしにち), and 二十四日 (only にじゅうよっか, never にじゅうよんにち or にじゅうしにち).
Otherwise, other dates from "the eleventh" through to "the thirty-first" are read with the expected 音読【おんよ】み: 十一日【じゅういちにち】、二十五日【にじゅうごにち】、三十一日【さんじゅういちにち】.