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Unfortunately, at that age I think it's really common to have short naps. Sounds like you're doing everything "right," and providing her a good environment to sleep in. Newborns just don't get the memo sometimes. I would either deal with the short naps until they lengthen on their own, or have your mom hold her. Or, she could hold her for one or two good naps, and try to put her down for the others.
Personally, we (me, DH or grandma) did hold LO for his naps until he was like 5 months old. It sucked being holed up in a dark room for hours all day, but now that LO's 2, I wish he would sleep on me sometimes!

When I transitioned my son to sleeping on his own in his crib I transferred my scent onto his crib sheets.

He was sleeping well on his back at night when we bed-shared, so I knew he could sleep on his back. The only difference was that bed smelled like us/ me/ breastmilk. I placed my used nursing bra in the crib all day and even rubbed it in a bit. Then, I started placing him in his crib. If he stirred and fussed too much I would get him out, snuggle and put him right back. It took two nights, but now he sleeps in his crib.

All that to say, maybe try making her bed smell like you and keep trying until the next feed or diaper change.

Have you talked to your mom about it? My mom watched LO for me for the first 1.5 years and honestly she was happy to have LO sleep on her! I was the one that wanted to get her more into a sleep routine.

Personally, I would be grateful to have a loving grandma watching LO at this age one on one rather than daycare where they do have to get on more of a sleep routine and learn to sleep on their own out of necessity. If your mom is okay with it, maybe let it be for now? Maybe look into a carrier for your mom so she can have LO sleeping on her while she gets things done?

Easier said than done though I was really about getting LO to sleep better & independently too! She was a little older when we did this but I would actually go and rock her back to sleep after short naps. About half the time I could put her back down after I rocked her back to sleep.

Just keep trying occasionally. I feel like keeping baby from being overtired is more important than napping in the "right place," so I didn't try for every nap. Eventually they get older and it gets easier. Somewhere in the 4-6 month range.

My LO napped on me until probably 3-4 months. I can't remember exactly when he transitioned to napping in his crib but it was definitely pay 10 weeks. He would take +2 hour naps on me but the second I laid him down, eyes would pop open. I'm pretty sure I made a post about it and everyone told me that it's normal. He started taking pretty consistent crib naps and now won't sleep on me ever. And it makes me sad!

Eta- there's a lot on the inter webs about the fourth trimester. Many people say that babies really aren't mentally or emotionally ready to be born until they are 3 months old but due to size and the way we give birth, we wouldn't be able to birth them. So the first 3 months of their lives, they need that contact from mama to adjust to the outside world. http://m.huffpost.com/uk/entry/9607642.html?edition=uk