I had 3 kids in 3 years, all with different sleep needs. My youngest gave up nap time at 2 years and 1 month, the oldest went until almost 3 years old. Here's what worked for us with all three, but obviously you need to find what you are comfortable with:
1. Nap time became "quiet time". We used the word quiet time with our children and explained what we expected. At 1:00 the child went into his/her room and could play quiet games or read by themselves. They did not have to take a nap, they just had to stay in their room. At 2:30 (that's what worked for us, but 3:00 would have been good, too), quiet time was over.
2. If the child was really tired, they would put themselves to bed for a nap. And yes, some days we found them asleep on the floor, surrounded by their toys!
3. If the child was asleep at the end of quiet time, then we would give them an extra 15 minutes and then wake them up to stay on schedule.
4. Bedtime became earlier at 7:00 pm, as there was no nap during the day. We found that with just quiet time, they were ready for bed at 7:00 and off to sleep quickly.
5. It took each child about 2 weeks of the "5pm crankies" to adjust to an earlier bedtime and no napping. We really tried to stick with the schedule during the transition to help their bodies adjust to an earlier time schedule. We also kept their meal/snack times very consistent, and that helped a lot too!
Don't feel guilty at all about your daughter spending time alone during her quiet time. It gives her a break from all the stimulation around her, and teaches her that she can entertain herself without you -- some thing that comes in very handy as she gets older.
Hang in there -- your daughter is lucky to have a mom who cares so much about her.