Yes, I would write a strongly worded letter. I think Park Plaza is now owned or controlled by some chain, possibly Starwood, though I would actually start with the manager.

Even if she really was going to bring you an ice pack immediately, she shouldn't have taken away the original ice and had you wait iceless until she got back. That's pretty incompetent for someone whose job is courtesy and helpfulness. The fact that she took your ice away and then never brought more is ridiculous.