Consider:
It is the winter, if you pull the plug on her now, which you are entitled to do, then you will lose a months rent plus utilities plus your efforts of lining up another tenant.
She will certainly do this to you as well on the back end of her lease, you can pretty much count on that. You will lose the last months rent which you may or may not be able to get a small claims judgement against her for.
So it becomes a question of whether you want to take the 1 month rent hit now or later.
This is what I'd do (and have done in similar situations):
I'd take her now but only make the lease 6 months so that you can get it in-sync with the prime renting season going forward and get a better read on this tenant. Tell her that you want to get your lease in-sync with the summer rental season (that is your privilege) and you will renew it with her in late June for 8/1/14 - 7/31/15 assuming there are no problems (late payments, problems discovered in June walk thru, excessive drama etc).
I would also go back and take a hard look at the data that she provided on the application so that you have a better chance of finding her after she leaves in case you have to take her to small claims or housing court. Get good information on the Emergency Contacts (relatives especially). It is far easier to get this info now then after they have moved in.
This does a few things for you:
1) It lets the tenant know that you know how to run your business right from the beginning.
2) It gives you a 18 month tenant instead of a 12 month tenant as renewal will most likely happen unless she is terrible.
3) It gives you the information that you need upfront for when you have to chase her for the last months rent.
4) You get your rental unit in-sync with the prime rental season going forward.