I've looked at your saved game. It is certainly possible for you to crush the Mongols in 110 turns. However, in order to do so, you will have to either commit to total war or start delivering freight to dramatically increase your income.
Do you know about Incremental Rush Building (IRB)? If you want to rush a military unit, the cheapest way is to switch to the lowest cost unit (above your current shield accumulation), rush that, then switch to the next highest unit, rush that and so on until you get the unit you want.
It would be a good idea to rush some military units in the cities near Washington to fight the Mongols. Use Howitzers, tanks and Mech Infantry (change from the battleship and helicopter at the very least).
Put together a railroad network. You have very few engineers and you are using them for irrigation and farmland. Roads and railroads are usually (IMO) a better use for engineers than irrigation and especially farmland. With a railroad network and some Transport ships, you can move your army very quickly to where it is needed. You can crush an enemy in one region and then quickly move all those troops to fight elsewhere.
You seem to be building a lot of Recycling Centers. You don't need them in cities with 10 shields; you might want Mass Transit if the city has a large population.
You are building Libraries, Universities and Research Centers. Unless you plan to research a lot of future technologies, you don't need them. You should also sell off your existing science structures (perhaps completing the tech tree first -- although it may be best not to so that the Mongols can't steal superconductor of fusion power).
Aside from the above remarks, some miscellaneous worker allocations and a few cities that could use harbors, you seem to be running your civilization reasonably well, with the following exception.
TRADE. You can get lots of income by delivering Freight (and earlier in the game Caravans) to proper cities. They give you an immediate payment in Gold (and an equal amount of science) as well as extra trade arrows in your cities.
Caravan bonuses depend on:
The total number of trade arrows in both cities (the more the better)
Distance between source city and destination city (the farther the better).
If the cities are on the same island (2 different islands doubles payment).
If the cities are from different civs (foreign delivery doubles payment).
If the commodity is demanded (x2 to x4 bonus, depending on commodity).
If the cities have superhighways (+50% for each city with superhighway).
If both cities have airports (+100% if on different islands, I think +50% if they are on the same island).
If you want to go for total war, begin by building some more engineers and putting everyone on railroad duty. Get some transports to ferry units between cities. Then start rushing military units (mostly howitzers and mech inf; with other units when you have a specific need for them), from cities with barracks if possible. Get a decent number of spies as well.