Are you trying to do an OTA Update through the Settings app, or are you trying to apply the updates using iTunes?
Try using iTunes from a computer to update the phone if you're not already. Needs less free space and is often a bit more reliable.
What is the error? Typically if you search the exact error you will come across an article that Apple Support addresses and may give you steps to fix the issue. But first identify the error that you're receiving.
Try putting your phone into DFU (recovery) mode and then do and update through there. Make sure to back up your phone first using iCloud or iTunes, just in case you hit the "Restore" option rather than the "Update" button.
I would also reinstall iTunes first before doing this, and making sure the phone is able to be synced. It proves that the connection to the phone is good.
Jcrook, I want your device updated. I want you to be able to take advantage of the new software for the device. Were you able to do a recovery mode as suggested by Smith6612. Have you tried a different computer and does it give the same error?
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