You won't get it decent if the dogs are going to use it. It sounds like 2 dogs is more than the balance your lawn can handle. Putting effort in is likely to be wasted.Adapt the lawn area to suit the situation... mulch, bark, stone chips, astroturf etc would be better.
I'm in the same boat...it's lovely in the warm summer months, but come autumn and winter it turns into a bog. We've had a load of topsoil replaced there to fix compaction issues and while the borders are now doing great, the grass can't hold up. We have three dogs by the way...I'm thinking we will rework the area, sans grass, as it just doesn't work there. We have another area of grass which is fine, not amazing during the rainy months, but liveable so we'll just have that be our little bit of lawn. I think as the others have said it's better to find something that works with the dogs.