Have you tried a restore through DFU mode? Open iTunes and connect the iPhone to your Mac. Press and hold the Home button and the Sleep/Wake button at the same time. After exactly 10 seconds release the Sleep/Wake button. Continue holding the home button until you iTunes pops up a message telling you that it has detected an iPhone in recovery mode. After that just click restore.
I suggest doing a factory reset through recovery mode, you can access recovery mode by powering off the device fully, then press and hold both volume keys and the power button for a few seconds until the screen flickers. This should take you into recovery mode. Navigate the menu with your volume keys and highlight "factory reset/wipe all data" or something similar and delete it with power. Then navigate to "yes" and press power again. This should reset your device to out of the box state.
Try doing a factory reset from recovery mode, turn your device off then press and hold the Volume Down + Home + Power button for a couple of seconds after its completely powered off and it should take you into recovery mode after a few seconds. Navigate the recovery menu with the volume keys and delete an option with the power button.
Go into recovery mode, and try a factory reset that way. Power off your device. Boot in recovery mode by pressing and holding Volume Up, Volume Down and Power Button at the same time. Release all buttons when the Blu Studio 5.5 screen flickers and in a few seconds the Recovery screen will appear. If it asks to restore a backup say no. Do a fresh install.
Best bet would be replace the charging port, this requires disassembling the device and desoldering the old port, then soldering a new one, it's not as hard as people think as long as you have basic soldering skills. :]