You just cut off the wire just above the plug that go into the cigarette lighter socket in your car, this should leave you with a foot or two between the end of the wire and the black box ( the PSU ) on the wire.
Remove the outer black insulation on the end and that should reveal the 2 wires for power.
I cant recall if they are the normal red ( positive ) and black ( negative ) color, maybe someone else can fill in on that.
You can then solder or crimp the wires to the wire you have coming off the old hard wire kit, or you can get a new piggy back fuse connector and put that in the fuse box in place of the one from the old hard wire kit.
If you decide to cut the wire from the old hard wire kit you of course have to make sure there is no power on it when you cut the wire, but that is probably the most easy way to get going, and if you cut the wire near to the fuse box the old hard wire kit should still be usable for something else some day.
Unrelated but similar:
This fluctuation power from a sub par power source ( in my case a really long and poor quality USB cable ) was the reason i dident have a working rear camera the day a few years ago when i got rear ended.
But it was my own fault i had noticed that some times the camera dident start, but i was just too lazy to do something about it at the time ( winter and i dont have access to a garage )
I am pretty sure your old hard wire kit are causing you the same intermit problems, but using the factory PSU that came with the DC should fix you and the camera up really good.