You're going to need to clean all that gunk up before you proceed.

Sand lightly, then use naphtha or alcohol to remove the old flux residue. Get your traces cleaned up and see which ones are broken.

Solder to original through holes as much as possible. Jumper from the FET legs as necessary to another component leg on that trace. If you try to tack solder to a small trace, you might have more problems, and that's not the best solution anyway.

When you connected the coil backwards, you blew the FET =and= you blew the flipper coil diodes. Cut them off, and replace with 1N4004 diodes. Note the orientation as the band on the diode indicates polarity. VERY important.

You can not operate the game until both the FET and the coil diodes are replaced.

It's an easy repair with the right tools and experience...without them...well...consider a local more experienced pin-friend or a qualified repairman or send it to a qualified board repair guy...Rob, Clive, me.
--
Chris Hibler - CARGPB #31
http://www.Team-EM.com
http://webpages.charter.net/chibler/Pinball/index.htm
http://www.PinWiki.com - The new place for pinball repair info