Nowadays a lot of laptop batteries are internally housed in the laptop case and connected to the motherboard as they don’t want the user to be able to easily replace them as before when they were inserted into the laptop from the outside.
Replacement batteries are available online. Just search for Asus Q505UA-BI5T7 battery to get results for suppliers
I couldn’t find a guide for your particular model but have found this teardown video for an Asus Q504U which may be of some help. The battery is removed within the first 3 minutes
Be aware that even though the laptop is off, there is still power available at certain points on the motherboard (unless the battery is totally dead of course). Disconnect the battery as soon as it is accessible to minimize any electrical problems
Try not to touch anything that you shouldn’t, (wear a wriststrap connected to earth to avoid ESD damage) and do not short out anything when using tools etc. The laptop power button is not a power isolating button. It merely signals to the BIOS/OS what the user wants to happen. Think of the laptop as being in an extremely low power state when off and not totally disconnected from any power.
Good luck with the repair