My thoughts on your situation:
First - clean up your code and narrow the function of the code down to -just- showing a count based on the sensor input. Turn the wheel by hand or via a battery. Basically doing this will rule out anything else that might be causing issues.
You have what appears to be a simple encoder wheel printed on paper and either taped or glued to the encoder wheel supplied with the kit and motor; it doesn't appear like you have mounted that paper wheel perfectly flat on the supplied encoder wheel; it needs to be kept as perpendicular to your sensor as possible. In fact, printing it on heavier stock may be in order. Also, increase the darkness level of your printer as much as possible, and make sure you are printing with black ink only (not some fake combo like some printers use) - ideally, you should be using a black and white laser printer and toner, on the darkest and densest print setting. Even better would be to use flat (non-gloss) black and white paint (instead of printing).
Your sensor should be covered as much as possible, to keep out as much ambient light as possible (lots of IR in ambient light). Also, you might take a clue from IR remote controls, which PWM the LED (usually at around 38 KHz), and have a means to sense only 38 KHz modulated signals - also in an effort to cut out ambient IR from the environment.
Finally, IIRC, that kit has a slot on the chassis for the encoder wheel to pass thru (?) - and you are supposed to mount a slotted IR emitter-detector pair on the other side of it; something like this:
...although the gap on that sensor seems a bit wide for your application (you would need to replicate something like the circuit you provided a schematic for earlier; that particular sensor I linked to only has the LED and photo-transistor; none of the other parts are included and would be needed to make the sensor work properly with the Arduino).
That wheel is slotted to act as an encoder; no need to print anything out, etc - though you still might need to enclose the sensor more (and/or modulate it) to help reject ambient IR.