Also, I may need to take back what I said about needing to replace your peal power system entirely; look at some actual E-bikes and you'll see none of them rely solely on electric power. https://www.electricbike.com/currie-ezip/
This is pretty much the cheapest E-bike on the market. It runs - that's about it. You should probably end up spending a little bit more than this E-bike's price on your own bike, since you found an incredibly cheap motor. You will want to source a speed controller that can handle your needs, then spend basically all of the remainder of your budget on a battery that's as large as you can get in terms of Amp-hours, since that translates directly to your 'range' or how far you can go before the battery dies and you're back to pedal power. Speaking of, we may as well take this opportunity to talk about batteries.
The reason you shouldn't use a battery as cheap as the one on this $500 E-bike is because Lead Acid batteries (this is explained in the article I linked you to) are really awful when you need to use them for frequent discharging and recharging by more than 30% or so; they require replacing much sooner than any other battery. You can damage them by discharging too much similarly to Lithium batteries. Plus, whereas NiMH batteries have 1/4th the energy density of LiFePO4 but are also about a third the price, Lead Acid batteries have half the energy density of NiMH. Buying them also is hit or miss, as you might find a 13Ah NiMH like this: http://www.batteryspace.com/NiMH-Battery-Pack-24V-13Ah-With-Charging-/-Discharging-Terminals.aspx
which is safe to discharge at a rate of 25 amps, which you need. 13Ah is fair, though $300+ seems expensive, though you will actually need to pay this much for cells that have the discharge rate you need and the added capacity is a bonus.
That battery seems absurdly expensive when you at first glance see this: http://www.batteryspace.com/sealedleadacidbattery12v20ah240whs.aspx
This is a sub-$40 lead acid which you could use two of in series ($80) to supply 24V, and they each have 20Ah! almost double that of the NiMH! Except that they can't even remotely supply the continuous current you need. The clearly erroneous listing of a 300A max discharge rate should make you very suspicious. Looking at the data sheet we can confirm that the battery may discharge at a rate of 300 amps; for 5 whole seconds.
There is a chart in the data sheet which shows that this battery cannot supply anywhere near 350W of power for your motor over a time period of more than 15 minutes. It would likely do terrible damage to the battery to even put this level of strain on it, even split between two batteries in series. Basically, do not use this battery.
If you want to get a bit cheaper this could work: http://www.batteryspace.com/nimh-battery-pack-24v-10-ah-for-electric-bike-and-scooter-with-battery-and-charger-combo-options.aspx
This comes with a charger and has 10Ah for a total of around $250, which is decently priced. I would still recommend you go bigger if you can however. Let me know if that all made sense or if you need any more help dealing with purely electronics.