I have a vague knowledge of the City as my son went to uni there. As op have already said most rented accommodation is not pet friendly, especially in the cities. Leeds has a good public transport system, so you would maybe be able to look further afield. There is an area called Hyde Park, which has lots of rentals, it is near the uni, so a lot of them are student or exstudent lets, there is lots of open space, but I don't know if dogs are allowed off lead there, I assume they would be as Leeds council seem reasonably dog friendly.

Temple Newsome park is on the outskirts of Leeds, it is a very popular dog walking area, but the housing around it looks quite posh to me, so I think rentals would unlikely to be available in that area.

There is also roundhay park, another very large area where you can walk dogs, and I'm afraid that is the sum total of my knowledge. Good luck in finding somewhere.