If you want to bump those rents up, you can try a furnished medium-term rental, but depending on the specific area, that might be tougher only because traveling nurses and remote workers tend to want to stay in areas that are a little further along. 

Are there any companies out there that lease small multi families for professional/nursing monthly rentals?

People who need a short rental for work such as traveling nurses, people who have sold a house and are building a new one or still looking for one to buy etc.

Guess I’ll have to just get into my sexy nurse costume and wait it out!

Much better tenants [corporate, nurse, insurance etc] and you are in compliance with state laws of short term rentals not being allowed in ADU etc] This way you can probably pay most of your mortgage. ie: we get $2500+ per month for a small 1 bed in Orange County as a monthly furnished rental. 

He has asked me if a travel nurse can stay with him in the house while doing a 3-month contract in the area.

Travel nurse for a while, folks looking/finding home to buy, folks staying while building on land out in the county.

I have experience closing a loan for a traveling nurse and it was scrutinized by our underwriter but we were able to get it done.

Since you guys have a big hospital I would try to rent it out to travel nurses using furnished finder.

I carry a nursing home administrator license in the state of Missouri (I live in the KC Metro) and I wanted to put feelers out there to see if there's a need for that type of expertise for folks maybe in the business of building LTC facilities (ILF, ALF, SNF, etc.)?