I just did this a few weeks ago in a 10x6 coop I built. I got my linoleum from Lowes, along with glue. Laid down the linoleum first without any glue, and cut it to shape. Then i lifter half of it up, in your case a 5x10 area, and glued it down, then repeated the process on the other half. The instructions on the glue calls for a roller to be used to flatten it out, mmmmmm my roller is a bit on the big side at 600 lbs and 4 feet long lol, so i just did the best i could using my feet. When i was finished i installed ‘base boards’ in the form of 2x4s, and screwed them to the wall, that way the glue hold the linoleum down, and the base board stops the edges from lifting. I also caulked the baseboards to form a seal, Its not perfect, but considering its there to protect the wood only, and covered in wood shaving its fine.