I knew a girl once who had an apartment at 12th and Taylor - it's now called Still-Blaine house. I thought it was nice then (haven't been inside one since the 80's); quaint may be a better adjective.
Today I see $790s gets you a big flat with arched entries, hardwood floors, sash windows, all appliances and utilities (gas, electric, water and trash) plus internet included. Pets OK. Sounds like it's probably higher than you wanted to pay, but there's a tiny cafe downstairs and it's next to a small park if you've got critters. Not in the state office zone and not in the depressed zone - it's kind of a micro-sized 'tweener neighborhood if you like downtown.
2-3 good restaurants in walking distance, the Walgreens at 10th/Topeka is the only 24-hour pharmacy in the city, and the YWCA is right down the street.
https://www.apartmentlist.com/ks/top...l-blaine-house
Good luck!