How can I replace the motherboard?

Depending on the cost of the board vs a complete system, it may not be worth it and may ultimately be cheaper to buy a used system and to use this one for parts. Consider this a practical lesson to keep drinks a reasonable distance away from your computer.

The other problem with these spills is in many cases, the keyboard also gets damaged. Sometimes you can save them, but most of the time they never work correctly again. That said, some of them do survive with the extent of the problem being a total keyboard cleanup being needed to get it back into shape again. Because it isn’t always known until you replace the board (unless it has obvious problems), be prepared to replace your palmrest assembly as well. So there are no surprises, it’s best to factor this into the total cost of the repair just in case it also needs to be replaced.

Anyway, you can find the board on eBay, along with the palmrest. The HMM can be found here, as there are no iFixit guides I can link you to on how to do this repair. The CPU is soldered, so you will need to check the part number of the new board against your broken one if you want to retain the exact CPU you already have. That said, you can always buy a board with a better one if you want to upgrade it while you are there, but the cooling system may also need to be upgraded. It’s less likely on this model since yours only has 2 cooler options (IGP/GPU) but if you decide to buy a dual GPU board and you have the IGP one, you will need the dual GPU cooler. These part numbers are in the HMM.