iPhone 5S error 4013

Error 4013 can be caused by NAND issues. Because your device was water damaged, the logical thing to do first is to have it properly decontaminated. Water tends to seep under IC's and can cause corrosion, shorts etc.

You could try the following yourself:

• Open your phone and remove the logic board (follow this guide)
• Inspect the logic board, especially around the connectors and look for corrosion.
• Inspect both sides of the board. Unfortunately, 80% of the board is covered in shields. That's usually where the damage is occurring.
• Put your board in a container with >90% isopropyl alcohol and let it sit for a while.
• Use a soft brush, like a toothbrush and lightly brush away any corrosion you see.
• Rinse in alcohol and repeat.
• Let it air dry for a day.
• Re-assemble and hope for the best.

You should also consider getting a new battery if it has swollen. Resist the temptation to !$$&*/pop it to let the gas out. A compromised Li-ion battery is a fire hazard. If the device appears to power up but behaves erratically, then use a tool like 3uTools to flash the firmware as it may be corrupted.

A professional repair shop that does water damage repair may be able to recover your phone or the data because they have access to pro-level ultrasonic baths and specialized cleaners as well as the skills to troubleshoot your board. Many shops have a no fix/no fee policy so you don't have to spend money to find out if the phone is fixable or not.