As stated above you will need a brake controller if you want to use the trailer brakes. A brake controller kit comes self contained with everything you need to wire the drake controller in except the round 7 plug itself.
As far as converting your flat 4 to round 7, check out auto parts stores, Tractor Supply type places or even the internet. You may be able to find an adaptor to go from flat 4 to round 7. That would save you any cutting or splicing, then all you would need to do is wire the controller in. If you cant find an adaptor thats ok, converting it from flat 4 to round 7 is simple and only takes a few minutes. No reason to go chasing wires if you have access to the rear of your flat 4 plug. as all the wires are color coded and match right up. Just buy a round 7 plug and cut and splice your existing flat 4 wires into it. You can do this and not use the trailer brakes just to get the trailer home and install the brake controller later.
When I installed my brake controller and round 7, I just cut and spliced wires from my existing flat 4 but I ran a T off of the wires so I now have both flat 4 and round 7 mounted under my bumper ready for whatever trailer Im using.