Shiliguri Corridor is called a Chicken's neck. In my opinion it is a neck of India with lot's of importance and lot's of weakness. Because, almost all of their high capacity roads, rails, power links, water canals etc. towards seven sisters are through this corridor.
All of these links has crossed several rivers, striking these links over the rivers will instantly isolate seven sister states from mainland India for a significantly longer period of time. DF-15 or DF-21C missiles of Chinese arsenal would be ideal for that, without even flying a jet over Himalayan, Chinese military can actually cut seven sisters states by pushing some buttons.
Once the links are cut off, Indian Military will try to use Bangladesh territory to reach heavy supply to seven sisters, this will be easily monitored by China by their spy network in Bangladesh. Also cutting few more river crossing bridges of India around Bangladesh will stop such effort.
When using Bangladesh territory is not a good idea, Indians will try to use heavy lift aircraft to move reinforcement, this can be countered by J-20 stealth fighters, without being able to see the enemy, and loosing assets at a very fast rate, Indian government would likely to feel losing ground under their feet.
India may also try to block Chinese trade vessels through Indian ocean, which will not be easy due to Chinese subs operating in that area, furthermore China Pakistan Economic Corridor will be a alternative trade route which can not be intercepted without getting involved with Pakistan.
Indian missile systems has severe accuracy problems, which would surely fail to cause much damage on Chinese side.
Indian air-force will be in big trouble operating near Himalayan areas specially when HQ-16, HQ-9, J-10, J-11 and J-20 will be operating from Tibetan plateau.
Air war over Himalayan will be very painful for India, with a jet to pilot ratio of 1:0.81 (less than one pilot for one jet) and aging fleet of soviet era jets, few SU-30MKI with huge radar cross section will be sitting ducks for Chinese air-force and SAM systems
One of the biggest problem for India will be to stop Maoist and other insurgents causing trouble inside Indian territory who are likely to get reinforcement from China during battle.
Calling America for help will be some effective, as they may help giving some satellite info and sophisticated technology to Indian military. But this is not fighting with ISIS, things will be much more difficult.
There is a good chance that America will make some trade in this opportunity by selling some very high cost weapons to India again which Indians will found to be no or less effective during battle.