You’ll probably have weekends (Saturdays/Sundays) off. You can spend Saturday roaming around the Old Fort City (Shahjahabad), which also contains India’s largest mosque.

Block off the following weekend for an overnight stay in Agra. Hire a car with a driver, drive after lunch on a Saturday, reach Agra in the evening, check in directly in one of the many hotels, wake up early the next day to drive to Taj Mahal and see it at dawn. Spend the whole morning of Sunday in the Taj plus a few hours visiting Red Fort nearby. Drive back to Delhi after lunch and reach Delhi in the evening.

On the third Saturday, visit the Indian National Museum, have lunch at Connaught Place, and check out other sights in central New Delhi. (I can give you a separate list.) You can also spend a quiet, meditative hour at the Baha’i Temple (also called the Lotus Temple).

On the last Saturday, prepare to leave India by shopping for stuff at the Tibetan Market or the old Ansal Plaza Mall (both in central New Delhi) or the new, modern malls in Saket or Gurgaon (south of New Delhi).

I’d recommend you visit Lutyen’s Delhi by walking around connaught place. Then take the metro from “rajiv chowk” to chawri bazaar. Try to get a rickshaw to show you around by taking the long route through Meena Bazaar and Parathe Wali via the Jama Masjid (India’s largest – stop and visit) (and eat at Karim’s – look that up on google, you’ll easily find it) to Chandni Chowk.

Also visit Lodhi Gardens, Humayun’s Tomb (for sure, especially if you don’t have time to go to Agra) and the Red Fort. Nizammudin’s tomb has qawwali on thursday nights.

For shopping, visit Dilli Haat on a weekend and browse handicraft and food stalls from all over the country.
Agra’s not inaccessible btw. Take the 7am train there, take the 630 pm train back.