I'm also making my first trip to India this year, at about the same time: last week of October and first week of November, 17 nights.
The itinerary Vuittons suggested sounds terrific, and I'll be staying at each of the hotels in that recommendation.
But one other option you might want to consider is the Aman India Journey, which is 2 nights at Aman New Delhi, 2 nights at Amanbagh, and 2 nights at Aman-i-khas, although you'd want to add at least one night to Amanbagh and Aman-i-khas to ensure a pace that balances all the off-the-beaten-path sites in with ample relaxation time. The Aman package includes all transportation between resorts, as well as breakfasts and some activities. And a tip: I booked the Aman India Journey through David O and Peirce & Leslie and got a rate that is at least $1000 less expensive than the published rate for the Aman India Journey. Plus, Peirce and Leslie guests are apparently frequently upgraded to the Pool Pavilions at Amanbagh.
One other advantage to the Aman India Journey for a short trip would be that you'd spend less time in the touristed cities and more time experiencing the local culture/color of less touristed sites. So particularly if you spend time in Delhi, that may be enough Indian city time for an 8/9 day trip, rather than both Delhi and Jaipur.