With extremely high dread of mongolian generals and experience of their elite troops, its gonna be close to impossible to fight all of the 6 stacks off with 1 army. Reinforcement is a bad idea as u cannot control them in M2tw, and even 1v1 can be a tough issue depending on your roster/general. I'm very surprised u held that long, as usually your lower-morale troops will rout as soon as their first cavalry charge.
The key is to use generals who can fight night battles, who can try to isolate and destroy one stack at a time. If you can, you can field huge stacks yourself - the AI will not attack 4 stacks of enemy army as they view it as far too strong for them to attack, and by using Zone of Control you can isolate mongol stacks and destroy them.

As for units, it sounds like you are playing English and Longbows are perfect for the job. You really don't want to charge at the mongols as your troops will suffer way too much damage (unless you are polish or some of thoese melee-intensive factions), and stakes at your flanks will take out enemy lancers. Your tactic is to lure their cavalry with light cavalry, while your longbowmen kill their horse archers (and they are very good at this). Have your general unit lead enemy cavalry across the line of fire of longbowmen to massacrethem, and when their HA and cavalry are neutralized use your cavalry to charge their infantry as they have almost no spears.
Using bridges to neutralize enemy cavalry is good too.

Lastly, if you don't have to fight mongols and they are not immediate threat, don't bother with them too much - you have to prepare yourself for the boubonic plague.