The trouble with most of the apartment complexes and rentals that you will find in N Lexington is that they are usually older, sometimes poorly maintained, and often in less-than-ideal neighborhoods. You could probably find a reasonably-priced apartment there, but I would be very careful about researching it well. Personally, I don't know of any that I would recommend but I'm also not that familar with the area. I do know that sometimes there are farms that have small cottages or farmhouses onsite that they will rent out. You might also want to check out the rental ads at kykernel.com - they are geared for the university community but sometimes you can find decent rental prices across town.
Georgetown does have decent apartments but you will usually pay more for them. If you ask around, there are often homes for rent that are targeted for co-op students and Toyota employees who are on short-term assignments in KY. Try looking in the local classifieds:
Georgetown News-Graphic: Classifieds
Most of Georgetown is fine but there are a few sketchy areas around the town.