The first thing to do imho is to restore partition type byte back to 0x83 using fdisk, but without reformatting!
If you didn't actually use the swap file (i.e. neither you nor init script run a swapon on it) you'll probably be able to recover the backup inode tables, mkswap only affects the very beginning of the partition. You should read these info/man pages: e2fsck, dumpe2fs, mke2fs (see -n option).
If you enabled the swap partition, then chances are most of your data has been overwritten several times, so any time spent would be wasted.
If the interested partition is a fat32, instead, you should probably google for a windows native recover app and cross your fingers...