G.B.
A toddler between the ages of 12 months and 24-30 months still needs a good nap during the day. Any older than that might be old enough to not need a nap. I hate that they get to that age and don't sleep anymore. I truly think that kids need to nap until they are in kindergarten. So, if she doesn't need a nap anymore you have to acknowledge this and let her stay up. She needs to have quiet time outside of their bedroom. Get the kiddo to sleep then do something with the older girl.
Sad as it is, it sounds like she is just not going to take a nap anymore and she's keeping the other girl from taking one.
If it were me I'd put her in a Mother's Day out or some other pre-school program that lasted through naptime for a while to see if they can get her back in the habit of sleeping during the day.