There is almost always a sort of checklist you can go through for dead equipment.
First conclude that your power supply is operating, and that it's going through the board perfectly fine. Check with a multimeter proper voltages, and if you want to be really fancy, use a scope to check the quality of the direct current (If there's too many variations in it, then it may hurt stuff)
Next, just do a sanity test. Look hard for anything, ANYTHING, that might be burnt, ripped corroded. Check continuity on anything that looks suspicious. You said that there was some corrosion, check every connection you can find around the corrosion for continuity.
At this point, remove anything that the system does not absolutely need to operate. Even if you can just get a fan spinning up, a beep, or something extra that you wouldn't otherwise, it's a start. You can go piece by piece to find what part is the fault if you can identify a state in which the system does operate.
If you reach this point, a logical choice has to be made. Reparation of any item requires time, and items that can't be fixed through traditional or easy means take a lot of time, sometimes not your own, and money. If you can truly say you need to have this thing fixed, then there are a few other courses of action in order of desparity. You may just want to buy a working laptop, or look elsewhere, as it may not be worth spending the resources to fix.
You can of course bring it to a repair shop, and tell them to repair it nevermind the cost. This is the least drastic, and most likely to work method. It will cost a lot of time, money, and effort, and you may have to take it from shop to shop depending on how well equipped the place is.
If you want to throw caution to the wind and go for a chance, you can always try to reflow the board. The idea is that some solder joints may have come loose, and by applying uniform heat to the board, you can reflow the solder, and repair those connections. The level of success is incredibly varied, and it may not even solve the problem at hand. If not done right it could result in permanent, irreparable damage to the machine.
You COULD do this in an oven, but I suggest if you do, you get a baking sheet to throw out afterwards so you don't get liquid tin (or even lead) in your next batch of oven chips. There are special reflow ovens available to people, often in maker spaces for a price. If you do end up doing this, read up on the procedure. Remove anything plastic from the main board. Anything that can be removed that is plastic should be removed.
If this fails then I have run out of ideas.
