Q: iPhone 5s can't connect to wifi and also not update to the latest software version

If your iPhone, iPad, or iPod touch won’t connect to a Wi-Fi network - Apple Support

Logically then your next step is a restore.

If you have literally tried everything there (all six steps, all four bullets under "Get more help", and all the nuances in each bullet and sentence; e.g., reset the router, update the router's firmware), and have done a restore as Johnathan Burger suggests, then the next two things to do are:

• Recovery.  If you can't update or restore your iPhone, iPad, or iPod touch - Apple Support
• Visit an Apple Store or Authorized Apple Service Center.

Below is a User Tip about Wi-Fi issues. Suggest starting from the top and working down. Maybe one of them will help.

1) Perform a Forced Restart:  Hold down the Home and Wake/Sleep buttons at the same time for about 15-20 seconds until the Apple logo appears. Allow device to restart.

2) Reset Network Settings:  Settings > General > Reset > Reset Network Settings. Rejoin the Network again.

3) Reboot Router/Modem:  Unplug from power for 2 minutes and plug again. Update Router Firmware (check manufacturer’s support website for a newer FW). Also try different bands (2.4 GHz and 5 GHz) and different bandwidths (20 MHz recommended for the 2.4 band).

4) Change to Google's DNS:  Settings > Wi-Fi > Click on the Network, Delete all numbers under DNS and enter 8.8.8.8 or alternatively 8.8.4.4

5) Disable device prioritization on the router if this feature is available. Also disable any VPN apps and test Wi-Fi again.

6) Determine if other Wi-Fi devices on network are working well (other iOS devices, Macs, PCs).

7) Try device at another Network, i.e., neighbors’, public coffee house, etc.

8) Backup and Restore the Device using iTunes (ask for additional for details if desired).

      https://support.apple.com/en-us/HT201252

9) Go to an Apple Store for hardware evaluation.

User Tip:  https://discussions.apple.com/docs/DOC-9892

2FR wrote:

 

Moreover, my phone doesn't connect to any wifi.

As the article I attached above states, in the third bullet under "Get more help":

"If it can't connect to any WiFi networks, contact Apple."

IMO you likely have a defective Wi-Fi chip or antenna, would be best to have it tested at an Apple Store. Good luck.