Here's a general idea of the wattage.
Assuming it's a tube, it's a T(x) lamp, where x = the diameter in 1/8" increments, so:
T5 = 5/8"
T8 = 8/8" = 1"
T12 = 12/8" = 1.5"
I'd guess it's a T8, so if it's 18" long, it'd be around 15W-18W, 24" about 17W-20W; it varies with brand, but as that was a whole 'starter aquarium set' it's probably on the low side on wattage.
You may also want to get a new lamp as those everything-kits tend to be on the lower end of the quality scale, but it's usable for a while.
Lamps are fairly cheap, from around $5 each to $20 each depending on brand, output, and bought online vs.
LFS.
T8 and T12 are
sometimes interchangeable, but I'd be cautious with a 'generic' type like you have, and just get the same size.
You'd want to get a good full spectrum lamp or a more specialized plant lamp.
DFS has spectrum plots for many of their lamp and some good general lighting articles to read so you have a better idea of what you need and want.
If you only want that Java Fern, then the one lamp is probably fine, but if you really want to get into live plants you will need to have more (sometimes much more) light.
-Edit-
pepetj :
The 'T' means 'Tube' and the number is 1/8ths of inches, as I mentioned, not 'standard socket', the rest of your decoding is correct though.