Restoring Iphone 5s

Both with a different cable and a different port? What type of computer are you using? If it is a desktop make sure you use a rear USB port. Do not connect through a hub. And you might try a forced restart of your iPhone while it is connected and see if that changes anything.

Try a different port and/or a different cable.

That isn't really a very solid period of time. I'm guessing your battery has dropped below 80% health or is failing. Click on the Contact Support link at the top right of this page for starting the service process.

How long ago did you replace the battery in that phone?