2011 13" Macbook Pro shows a green charging light but battery was draining and now it won't respond to SMC Reset

Unless you have had the battery replaced in the last few years it’s most likely dead. You can use a couple of different apps(Battery Health, Coconut Battery), or even the built in System Info screen to see the charge cycles and condition of the MacBook's battery.

I helped a friend with almost the same identical problem on a system of the same vintage, and it was the battery that was dead. No matter how many times he reset the SMC, it never helped.

As a side note, Apple recommends that you cycle the battery fully at lest once a month to avoid potential problems that can occur form charging the battery to soon in a battery discharge cycle. So if you are always plugging into power when you have only 30% of your battery drained, you could be shortening the life of the battery.