And please don't say "Solitude". I have been running around trying to find this guy in Solitude, but I just can't find him. Can someone give me directions within the town as to where he is?

It's a chick, and she's at the "Blue Palace" at the far side of town.

She is in Solit... ehh oh wait. She is at the Blue Palace, first floor. Just go all the way through Solitude and you'll find it. It's the last building.

She is sitting on the thrown up the stairs. She doesn't have the option for buying the house. Does anyone know how I would get it?

You need to make the city like you(do quests for her assistant).