40 VLS cells is not gonna do it. You need at least 64 and 8 more AshM units to be able to knock a destroyer class out.
When addressing how Nilgiri or any IN ship will do against type 55, once can't treat it as some MMA match where it's 1v1 in the ring. The IN as a whole will need to be looked at comprehensively, against what the PLAN can bring to the IOR, keeping the geography and chokepoints of the Indian Ocean in mind.
Many members have already explained how these ships are network centric, where all the ships can share sensor data and target tracking data in real time with each other. For example, if 3 nilgiri class ships are there in a wide area, they essentially act as one ship with 3 sensors at different locations, with a total of 24 Brahmos and 96 Barak 8.
Second factor is the location of the fight. It will most probably be near the Malacca strait if there's a war, so it's well within air cover and surveillance cover from the A&N islands. The Air power India can bring to this chokepoint is tremendous, and neither the PLAN nor PLAAF have an answer for that. The numerical advantage of the PLAN also diminishes due to this factor, since if they try to rush their ships into the IOR through Malacca, it will be easy pickings for our aircraft and submarines.
Let us assume that some PLAN ships make it through Malacca and come deeper into the IOR. All the relevant trade routes in IOR are close to our coast and are again within striking range of our aircraft. So the PLAN is stuck at the outer edges of the IOR outside our aircrafts reach, from where they can accomplish nothing against India, and will be running low on fuel and food anyway.
India needs to step up it's air and missile presence in A&N, and build large numbers of frigates and destroyers that are networked together and spread out, instead of fewer cruiser sized destroyers like type 55. SSNs would also be a huge benefit of course.