Then you could use any typical brushless ebike / scooter controller for it. If it just has three phase wires then you need a sensorless controller. If it has three phase wires and 5 (or 6) small sensor wires (hall sensors, plus possibly a speed sensor wire) then you can use a sensored controller for better startups from a stop.
You just need to define what you want the bike to do for you and how you want it to do that, so that you can pick a controller that has the right feature set.
Some are just a controller, no display, so they typically have no settings or anything you can change. They just do what they do the way they were factory set to do it. Some have a display that shows you various info (speed, battery remaining, etc) lets you change some settings, and possibly have multiple levels of assist, etc.
Some are PAS-capable (where how you operate the pedals controls the motor), and there are a number of ways those work, from the most common on/off (if pedals are moving, full power at the chosen assist level is provided) to cadence or even torque-based control, where either how fast or how hard you are pedalling controls the amount of motor assist.
Some are throttle-only, and some are throttle-controlled while you are pedalling, etc.
Then you can also define the amount of power you need in order to do the job you want under the conditions you have. Meaning, how fast you want to go, on what kind of terrain, under what wind / etc conditions, and how quickly you'd like to get started from a stop, etc., and how far you need it to go under worst-case conditions at the highest speed.
If none of that matters and all you want is to go without pedalling as hard...it would probably work just like it already is, once the battery is replaced or fixed.
If you need higher speed, you'll need higher voltage and higher power than the original provided. More power will heat the motor more; if it's like most, it can take a little more than "rated"; a lot more continuously might be a problem. A few MPH more than 20 is easy enough; say 25mph would probably take 750w or less, depends on conditions. More hills or more headwinds takes more, flatter roads with no winds takes less. Even tire pressure/etc matters.
Simulator at http://ebikes.ca/tools/simulator.html may help you see how it works and help you pick the size of controller and battery you'd need to do a particular job.