Can I retrieve data from a phone with a broken screen?

If the screen is the only thing that's broken, its entirely possible to retrieve the data. If its just pictures you want to save, you can simply connect it to your computer, and download them off of the hard drive. All of your contact information and the like is stored on the SIM card, which will likely be moved to your new phone. If it isn't, you can ask the store to copy it for you. Everything is still stored on the hard drive, so you haven't actually lost anything by breaking the screen, though you could consider replacing the screen. It sounds like you may have a broken back-light, but its not clear based on your question. In any case, best of luck.