Your fuse should not be getting hot. Sounds like the strip has failed or you have a short to ground.
LED light strips are inexpensive. They are just peal and stick and you connect a pair of wires. Just replace the strip and be done with it. You can even get multi-colored versions so you can have a nice disco party.
My stock LED light strip is at the end of the awning which requires the awning be fully open for the lights to work. I added a separate strip on the side of my rig right under the awning housing and wired it into the porch light. I use this strip all the time as the awning does not need to be extended.