You can cover the Golden Triangle in 6-7 days. For the rest of the holiday, I suggest a 3-4 day visit to Rishikesh, which is in the foothills of the Himalayas. It does not rain incessantly there nor is it so high up that there is a danger of landslides during the rains.

It is one of the best destinations in India for religious, spiritual, yoga and adventure tourism. Of course no adventure tourism during the monsoons.