Depending on your budget -- "palace on wheels" would be perfect giving your time constraints.
At least all of your travel issues have been taken care of and you can really relax and enjoy the trips...the Indian countryside is fascinating to look at while on a train.
This also would cover your hotel issues, domestic airline issues and everything else. Agra to Delhi is approx 1/2 day trip by car, so you land in DEL and fly out of DEL. Could not be easier !!