How do you repair a printer if an ink cartridge leaked ink all over the inside of the printer, leaving it dirty and now paper won't feed?

Printer cleaning sheets, perhaps.

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Actually, printers are so cheap nowadays.  I would advise you to get a new one.  You did not say what type of printer it was, but the things is, you will never get every nook and cranny cleaned on the inside and once you think you have, as soon as the gummed up ink dries up on moving parts, it is going to jam again and again.  Not worth the headache.  Look at reviews to see what printer will work best for your needs and go that route.

Even a properly aligned cartridge can leak if the contacts between it and the machine are dirty or clogged. Similar to before, open up the machine, remove the offending cartridge and wipe it down with an alcohol-based wipe – not water as this can cause damage. Then, do the same for the printer head.

You need to open up the machine and clean it with alcohol based wipes not water. Water can cause of damage. 

try to change the ink and then if you can clean, clean the feeder...... and it doesnt work.   BUY a new one with better Ink seater....

if its a cheapie throw it away

Impossible to answer because the lay out, rollers, feed trays, etc., are different for every make and model of printer.  There is no one answer to how to clean a printer.

Depending on how much access you have into the printer to be able to clean the rollers, etc., you may or may not be able to save it.  If you can get inside you clean it like you would anything else that has ink on it.  Just avoid getting any of the electronics wet.  And of course be sure it is unplugged before you do anything inside it.

it's dead, Jim.  throw it away and get a new one -- you'll  NEVER  get it clean

You don't. New printers start at $30. Time to buy another one.