If I understand correctly, it starts the boot process, connects to iTunes (when connected to a computer) but then immediately shuts off. Does it loop through this process or do you need to turn it on again?
From what I can tell, the battery is pretty much fully discharged and won't charge. That's why you see the charging icon briefly and then it shuts down. This could be caused by either a bad battery or a damaged charging circuit.
So there are a few things to isolate here:
- Do a good visual inspection of the logic board and the connectors. Look for dust or other impediments to a good connection and look for missing parts, especially near the battery connector. There are critical components there that are tiny (0.5 x 1 mil, or thousandths of an inch).
- You need to replace the battery with a known-good battery. Buying a battery does not make it known-good as there are many poor quality batteries on the market. iFixit sells good quality batteries but even they are not 100% (sorry guys!).
- If the battery doesn't solve your problem, then changing the Lightning Port is next. It is between the battery and the charger, so to speak, so it could be causing the issues. Again, there a lots of low-quality "small parts" on the market and buying it "new" does not mean it works.
- Are you are confident that you re-assembled the device properly (you mentioned the screws) ?
If all of that does not solve the problem, then you probably have a logic board failure that was caused by the impact that broke your screen. In that case, you will need to find an experienced micro-soldering repair shop to troubleshoot and repair the problem.