Finding the settling time of an oscillatory response
The first thing you should do is to define what you mean by settling time. Then the method for finding it will then be as simple, or as difficult, as the definition implies.

The two main definitions are:

a) The last time that the signal passes through some threshold

b) The time that the theoretical envelope of the signal passes through that threshold

With (b), you only have to solve for the decaying envelope, as you've done in your OP.

With (a), you need to take account of the detailed oscillatory behaviour of the output, which involves finding a few candidate cycles around the time that (b) gives the answer, and tracing through which one contains the actual time. 

While computing (a) might be an interesting mathematical exercise, there can't be many practical applications that really need anything better than (b).