This motor would work, although I would recommend learning how to solder personally. The motor itself is not actually broken, the wire just came off, and you could easily strip the rubber off and solder the wire back into place.
It is an investment in time and money to learn to solder, but if you're interesting in getting into electronics soldering is a necessity.
If you are not up for that, then you could just buy a cheap replacement online or at a Radioshack too. The motor is basically just a cheap toy DC motor.
But you can also buy solder in a tube, it's a sort of metallic glue. I've never used it but it would probably be ok for that purpose.
Those are usually known as "toy motors", the exact spec will not matter for starter kit kind of use. I don't know what voltage the starter kit uses- does it power stuff from a battery?- Just get a toy motor of the same voltage specs. Here's a $2 one at adafruit.
Lesson to learn here is that the wires should be supported somehow to stop flexing at the joint. Can be a good idea to loop them gently back over the motor and strap them to the body with a cable tie.