I've only ever got this far as England but they have lovely heavy infantry and archers. With both Mongols and Timurids I played a waiting game giving territory for time to build up units. I then blockaded river crossings to contain their armies away from my cities and used assassins to eliminate their lesser generals. I chose to engage only their weaker armies on the periphery and avoided engaging their multiple stacks.

I eventually took on the Mongols with lots of cavalry (as England I'm not sure this was the brightest move) and the Timurids with high quality archers and infantry. As you've mentioned the Elephants are like medieval tanks and your only hope is too sap their morale until they rampage through their own lines. I mainly used flaming arrows and heavy cavalry charges to achieve this.

In the end this strategy was time consuming and expensive, preventing me from exploiting the Americas until the very end of the game. On the positive side they offer the best battles of the entire game, giving it a real kick just as things seem to be getting too easy.