Headlight restoration kit should work fine, but it probably doesnt come with enough sandpaper. Buy some more from either home depot or amazon. Personally I would buy a pack of sandpaper that has a big range. Start with 500 like the kit has, but then work your way up to like 2000 before you start moving on to foam polishing.
If you have a buffer/polisher then you can use finishing level foam pads from that as well for your final polish step. They would be a little more fine than the foam the 3M kit comes with.
Once you are done, use something like these:
https://www.amazon.com/Chemical-Guys-GA ... 06ZYGJ4W8/
https://www.amazon.com/Meguiars-G17804- ... 01M4RVVX6/
Spray/wipe it on your headlights every couple months right after a wash to keep them looking nice.