I am not sure why you want a "feminine equivalent." The trend these days is to use gender neutral terms (flight attendant instead of either steward or stewardess).
If you must stipulate the gender, you can say a female intern. Woman intern would be less likely. And since girl, when applied to a female over 17, is seen as sexist by many but not all, I'd stay clear of that. Likewise intern(n)ess would come across as sexist, like stewardess. (Some people even avoid using actress.)
Note you could also just invent the word "homewoman". Presumably your context would show you are trying to use a "feminine equivalent" to houseman. English is not limited to what's in dictionaries, but to those who write in the language. That's why neologisms, or new words & expressions, are always possible.
Words ending with -man are not necessarily "masculine words". For instance, Fortune magazine reports:
Exclusive: From AOL Exec to Antarctic Explorer, Meet National Geographic's First Female Chairman
This usage also leads support to the term female intern and not woman intern.
Indeed, Oxford dictionary reports that such "gender-neutral terns" as chair and chairperson are "far less common than chairman."
Last, simply FYI (for your information), instead of houseman you can use house officer:
A qualified doctor practising under supervision in hospital in the first year after graduation.
Both houseman and house officer are "British" or "chiefly British."