Sometimes the AI can get a bit stuck in regions, I don't recall it happening to an entire horde like that though.
Firstly, I think you'd need to have one of your armies on the bridge for it to spark a bridge battle. You might find you have better luck there if you put a relatively weak force on the bridge - still likely to win or cause significant damage but less daunting for the enemy. Currently to attack they'd need to move onto the bridge and have one of their stacks against two of yours (albeit possibly in an open field battle).
In an open battle you've got to beware the amount of horse units and versatile archers but they're not crazily scary, especially if you're prepared to exploit stakes to their full advantage. I'd put together a stack almost entirely of longbows (with a decent general) backed up by a second stack with other units in, deploy a horrid array of spikes in a defensive position then pull back (so horse archers will need to be in the region of the spikes to shoot at your troops) and wait for the Mongols to attack - even though you'd be the attacker the AI tends not to be dull even if being dull would make tactical sense. Their mounted units will fall foul of the spikes and, especially if their general dies, the infantry won't like concentrated arrow fire. If you've got gunpowder then that's even better to get them to run
The Timurids are the difficult ones...
If neither of them provide the challenge you're looking for there's always the
Mongolphobia mod too