I’d say the Oberoi Delhi is the best option in Delhi, and while Delhi is chaotic, it has plenty to do and see to fill 4-5 nights itself, especially if you visit Agra as a day trip (a long day trip - perhaps you can swing 2 days in Agra to see the other sights in Agra, Fatehpur Sikri, and Vrindavan/Mathura)? The Oberoi Amarvilas is the only true luxury option in Agra.

Jaipur is great, but 4 days is a fair bit of time (2-3 is more common) - do you prefer spending more time at destinations just for cities or to explore in depth and travel slower?

For a third destination I think Udaipur is good - it’s less chaotic for an Indian city so you can relax with enough to see for a few days itself and nearby with Ranakpur and Kumbhalgarh. There are a few good options here to stay with Oberoi and Raffles and others, but many people spend 3-4 nights here.

I guess that takes you to 4 (Delhi), 2 (Agra), 3 (Jaipur), and 3 (Udaipur) for 12 days with the option of spending the time in Agra in Delhi and / or Udaipur instead?

Other options in the area (depending on when you are visiting) to replace Udaipur would be Ranthambore for a tiger safari (note your odds of spotting are lower in the wet season) - Oberoi and Aman the best for this, the Himalayan foothills (Ananda is a yoga and spa resort with natural beauty - not sure if too many other ultra-luxe options but there are many very good options in that area), or maybe rural Rajasthan maybe Six Senses Fort Barwara or Amanbagh to relax?

South India has similar regions but for a 10-12 day trip I’d suggest staying up North and these would all be close-ish together.