I looked up public transport and apparently the Yorkshire Coastliner # 840 towards Whitby stops nearby. That's the stop: Kiddal, York Road Park House farm (Stop 45013224) (on York Road) and from there it takes about 10mins walk to the Dogs Trust.
I think there are other animal shelters nearer to Leeds. Have to google...
Sorry, things are a bit mental at work. Hard to keep up with booking and people etc.
As far as I can tell there is no Dogs Shelter/rescue in Leeds itself. It's good for walking the dogs and space to have it a bit outside. There is the Yorkshire Animal shelter but that's just as outside as the Dogs Trust. It's by the airport and they don't actually have dogs or they didn't use to.
The places in Leeds are for smaller animals cats etc e.g. camp nibble www.campnibble.com is for rabbits, hamsters etc.
If we take the Coastliner we should be alright getting to the Dogs Trust, don't you think?