Yes,it's cold out there and the Broad Street run is over 2 1/2 months away, so it seems like you've got forever to get ready. But it's important to start getting ready for the May 1 event now if you're physically able.
Of course, just how much you should train depends on where you are. If you run regularly, say 30-40 miles per week, then it will not be a big deal to build up to a 10-mile long run at least once a week. And once you're there, if you can build to two or three 15-milers before Broad Street, you'll really feel confident about competing for 10 miles.
If you're just getting started, it's vital to work your way up gradually. It will make the coming weeks easier on you mentally and physically. Too much too soon, and you feel like you never want to do this mentally, because everything hurts physically.
Sometimes a runner will ask if he or she can improve their time in a long race, such as Broad Street. Of course you can, it just depends on how much you want to train. If you have the time and motivation to get serious about training, it would be a big benefit to compete in 5Ks by at least mid-March, then look for a 10K or maybe something longer in early April. Even if you find a half-marathon in early April, just so you can compete a little and be happy to finish, that will really help for those 10 miles down Broad Street.
Another big question is if you've trained hard, when do you start resting for the big race? It's important not to shorten the mileage or intensity too soon, or your body and mind will be in for a little shock. However, you don't want to be super intense the week of race, or you just won't have the energy to go faster when you want to. So, I usually go for somewhere in between. If the big race is on a Sunday, I think you're still OK to do a long run on the Monday before, then a short, but difficult workout on Tuesday. But then you need to start pulling back, Wednesday, then just do some really light stuff Thursday, Friday and Saturday.
The other factors that play into race performance are of course what you eat, how much sleep you get and how much you drink. Remember hydration should be a habit, not a one-day binge.
So, start preparing now mentally and physically. You're body will thank you come May 1.

Here's a quote from the late great Steve Prefontaine that may help you get out the door on these cold mornings:
"You have to wonder at times what you're doing out there. Over the years, I've given myself a thousand reasons to keep running, but it always comes back to where it started. It comes down to self-satisfaction and a sense of achievement."