Balancing a large 18650 battery bank
You certainly can have some fun doing this project. Dead Laptop batteries still might have few fairly good cells to harvest. Keep in mind however that the internal cells don't have any protection, so be careful not to short anything at a risk of "rapid venting with fire". 

There are some hackers who do this kind of projects and offer their advice, like this one. In most cases take their advice at your own risk, since usually their methods are not backed up by any test nor reliability/safety research. For example, they would boldly connect 18650 cells in parallel in large numbers without any reservations, which doesn't make them overly reliable and long lasting. 

For a overview of some challenges in the area, I just found this nice article, which I highly recommend for a start.

Finally, I am afraid you can't just "shell out money for real batteries" and expect you EV bike to fly. One battery doesn't make a EV. You would need a full EV solution for power management, braking recuperation, etc, all sorts of fine engineering.