The info. on my phone is very important and my phone is broke and needs to be fixed.

If the computer is not recognizing that your iPhone is connected then you have more that screen damage and Apple will only do an out-of-warranty replacement of your iPhone. Your warranty is already gone so if the data is that important to you (and if it is why aren't you backing up to iCloud or your computer nightly) find a competent 3rd party repair shop that has the equipment to do board level soldering. They may be able to fix your phone where you can continue to use it or at least long enough to back it up. By doing this you probably will not be able to get future support from Apple on that device, including out-of-warranty replacement.

I would suggest you start a backup plan.

Have your device inspected and repaired, telling Apple Specialists that you don't have a recent backup.

They could try, without any garantee, to keep your data. If something goes wrong... No Backup = No recovery.

Make an appointment at an Apple Store or AASP: Apple – Location Selector

Regards

Giulio

There are services out there and for a few hundred dollars can and do successfully pull data off broken phones (as well as fix them). The Genius Bar and Apple Authorized service centers will guarantee not to void your warranty or risk you losing your trade-in credit if you upgrade. These non-authorized repair shops are an option if the data is that important. As with any option, do the research and find those that are highly regarded (and don't be afraid to look out-of-state). I had great luck with a specific facility that I found after searching for a few minutes. I'm sorry for your situation and wish you the best!

I do apologize for my poor wording - I was trying to illustrate that the Genius Bar and the Apple Authorized would not void a warranty where 3rd party would and negate the possibility for trade-in. Additionally, I didn't see your comment until after I had posted mine as the forum layout doesn't always show all comments unless you explicitly tell it to. I am paid by a 3rd party and love my job - I work as a database administrator and couldn't fix a device to save my life. I wanted to join this community as a way to help others and give feedback as I do enjoy my Apple devices and am fairly savvy.

Again, I apologize for the repeat and misunderstanding.

How can a 3rd party guarantee that Apple won’t void a warranty? Does she own Apple now?

I already said all this why are you repeating it? Do you get paid for your posts by a 3rd party?