You might do better with a carpet cleaner, but depending on the quality of the carpet if you have a wet and dry vacuum cleaner you could use that instead. The issue is that every time you add water to clean it, it is driving some of the wee-stained water yet further into the carpet pile and backing. So, alternately giving it some cleaning water, then immediately extracting this either with a carpe cleaning machine or a wet n dry vac will start to bring out the wee.
Alternatively use lots of dry towels and trample them into the affected area each time you wet it with cleaner and water, so the water and wee is soaked up into the towels. I mean really A LOT of towels. Give these a good hot wash immediatley after and no harm will come.