Is it ok to ask for an expensive item since I don't get paid for caregiving?

My grandma lives with me and my mom in my mom's house. I am not getting paid to be Grandma's caregiver and Grandma gives us $300 per month to cover her expenses (groceries, incontinence supplies, her heating costs, crossword puzzles, medications, etc.).

Because my desktop computer has to be upstairs and Grandma is downstairs, I often have to spend time away from her. There are things I need to do on the computer and because my internet is insanely slow, it takes up a lot more time than what it should. I do not have any mobile devices other than an ancient flip phone.

Additionally, I often get bored sitting in the living room with Grandma. Her game shows bore me because I've seen them all before (some I've seen many times before) and I can't watch anything I like because it's either inappropriate for her or she can't follow the plot. I've tried working puzzles (crossword, sudoku, etc.) but I end up getting agitated after a while. Often times, Grandma becomes annoying, particularly after supper. Either she'll talk non-stop (annoying if you're trying to work a hard puzzle) or she'll make annoying sounds. I've often stomped out of the room in total frustration. I have no idea if she's aware she's making those sounds or not. I can't believe anything she says.

I'll also point out that I often hear Grandma making sounds when I'm upstairs using the computer. The floors and walls are so thin. I can hear her coughing and crying/whimpering (not sure what that's about). It's very distracting and it makes it hard to concentrate. I often get annoyed because I can't put on headphones or anything to block the sound because I need to be able to hear her in case she calls me. I've tried having her use an intercom, but she won't use it. She wouldn't use the loud bell either.

Since I'm not getting paid and I had no idea Grandma was so annoying (she did not make annoying sounds before she moved in), I'd really like it if Grandma would buy me a laptop computer so that I can spend more time with her without me being bored and frustrated. I figure that if I'm sitting with her using the laptop and she starts making noises, I can just put on headphones. If she needs me and I can't hear her, she can just wave at me. Plus, with a laptop, I can show her family pictures, get her to help me with our family tree and history (using websites) and maybe help her play an online card game with her sister. I think it would make a huge difference in Grandma's happiness because I'd be spending a lot more time with her - she HATES being alone. I figure I would spend anywhere from four to eight hours a day with her versus the 20 minutes to three hours she usually gets.

Would it be wrong for me to ask the POA for a laptop computer? It doesn't have to be fancy. I figure it will cost between $400-$700 depending on a number of things. If Grandma didn't live with me, I would have no use for a laptop, so I don't think I should be the one to pay for it. Opinions?