It sounds like you've got your hands full there, with plenty to do.
What needs doing on the bike to make it road legal? I don't know KX125s, but I'm sure someone will be along in a minute to offer some advice.
As far as getting your licence is concerned, there are 3 main options:
1) Do the test on a bike with <125cc and power of up to 14.6 bhp and get an A1 licence, which will limit you to small bikes forever, unless you do another test
2) Do the test on a bike with 120-125cc which is capable of 60 mph and get a category A licence. After that, you will be limited to two years of riding a bike of up to 33 bhp and a power/weight ratio not exceeding 0.2 bhp/kg, after which you can ride any size of bike that you can insure
3) If you're over 21, do the Direct Access test on a 46 bhp bike, and immediately ride any bike that the insurers will let you get away with.
If you reach 21 and are in the 2 year probationary period, you can also do Accelerated Access, which is basically doing the Direct Access test, and passing that again allows you to ride any size of bike.