Personally, I would take her out and be done with it. You've given it over 1.5hrs which in some rule books is the length of time they say to keep them in bed for. She's been happy enough to play quietly in her room and therefore giving her some reasonably unstimulated quiet time which as they get older is all they need if they are done with the napping.

She may get tired a lot earlier tonight, but if it was me, I'd keep her up to her normal bedtime (give or take 30min), do normal routine at bedtime and I'd imagine she'd be off in a tick and get a good nights rest.

You may find towards the evening she'll get grumpy and a little harder to handle but there's nothing much you can do about making her have a sleep, sometimes it just doesn't happen especially seeing as she's getting close to 2. Sleep cycles change all the time.

Good luck.