IDK who needs to hear this, but diapers make amazing ice packs. I paid like $7 for 48 newborn size, filled 3 of them with water and froze them overnight. They stayed somewhat curved, so they fit my shoulder perfectly, but you could probably flatten them out if that's the shape you need. Puppy pads would probably work too.
I've used the same one since 7 this morning and it's still about halfway frozen, still cold enough to be effective, with no mess. I think next time I'll use an alcohol/water mix for flexibility. Parents probably been knew this trick but I thought I'd put it out there.
I've been using an old T shirt to wrap it in, but I think those little sacks that new sheets come in would make a perfect cover

















