I am doing this all from memory as I have not been brewing in a decade or so, but this is how I have been carbonating some sodas using a "carbonator" and 2 liter bottle, should work with a keg also.
Once cold it will take a lot more carbonation, but not necessarily fast. To force carbonate fast get the kegs really cold. Put in a bucket of ice, water and salt. Let sit for a day or so until they are cold through. Put a head of pressure on the keg, then roll or shake up the keg. The head of pressure will go away as the CO2 is absorbed into the beer. Put some more pressure in the keg, shake, roll or whatever. Let sit for a few minutes to settle down and sample, repeat if you want more carbonation.
My guess would be 30 minutes of effort should get your beer carbonated.