I take it the smoke is coming out the tailpipe when you crank the engine, even though the engine is not starting or, at least, not running to the point where the car can be driven. It sounds like you might have 'cooked' the engine as old school mechanics would put it. When the radiator started leaking, allowing the coolant to escape, the engine overheated and 'cooked'. Which means the severe heat inside the engine damaged something sufficiently that the engine now won’t run.

You could be dealing with a blown head gasket or a warped cylinder head, or even damage to the pistons. The smoke you’re seeing when you crank the engine could be oil smoke that’s a result of fired piston rings due to the overheating episode. The first thing to do is let a mechanic run a few tests on the engine to determine whether it’s retrievable or junk. From there, you can make an informed decision on whether to fix the car or cut your losses and simply walk away. If the engine is damaged beyond help, a second-hand engine from a wrecked Tiida might be an option if you decide you want to keep the car.

The check-engine light on any modern, fuel-injected car is there to let you know there’s something amiss with how something within the driveline is working. However, precisely what that thing is could be one (or more) of about a thousand things. The absolute best advice is to take the car to a workshop and have it electronically scanned. The onboard computer (which is what triggers the check-engine light) should have recorded what was at fault and should then be able to tell the workshop’s computer where to start looking.

Until you have this scan performed, you won’t have a clue what the problem is, and the causes could be anything from a dud oxygen sensor, to a stretched timing chain, and a dead spark plug wire to a blocked fuel-injector.

You may have a damaged or worn transmission that is simply not operating properly in a mechanical sense. But it’s also worth checking the level of the transmission fluid because low fluid is a common cause of this sort of problem. Without sufficient fluid and, therefore pressure and volume, the gearbox can’t operate properly, with symptoms including slurred shifts, transmission slippage, clunky or noisy operation and slow gear selection.