@20GTX,
Why bother? If the staff are willing to get her, what's the problem?
Otherwise, you can simply say to your cousin what you said here (without all your "no advice needed, stick to the topic comments) - you want her to have a damn phone and what will it take? Or, since she's probably annoyed with you by now ( not responding), maybe just leave it?
Why include info which is irrelevant and you comment on, such as:
"Cousin (now power of attorney) moved them to Hawaii. Stayed years. (no advice needed about this, please stick to the topic)"
and
"My mom is now in a NC home for people with memory problems for the last year. (no advice needed about this, please stick to the topic)"
Waste of space and distracting.
The simple question is always the best - My POA Cousin hasn't provide mom with a working phone. How can I politely prod her along?
Answer: No idea. Don't know you, your mom, your cousin (or your relationship with her), the nursing staff, how far along your mother is in her dementia (would she know it's you?)...