You have a PC with Windows XP operating system. A wireless keyboard & mouse. As nearly as you can tell everything is working, except you cannot get input from your keyboard to show up on your screen.
You tried changing batteries in your keyboard and tried to re-sync keyboard. You can try removing it's driver from device manager in control panel. Click your "Start" button then select "Control Panel".
Choose "System" then "Hardware". Click a "Device Manager" button. You can right click keyboard and select "Uninstall".
Once keyboard is uninstalled then you can click on a "Scan for hardware changes" icon in device manager. Windows should reload a driver for your keyboard. Make sure to uninstall your wireless keyboard device.
There should be a "Scan for hardware changes" icon towards top of that Device Manager window. It looks like a small computer monitor with an hour glass. If re-installing your driver doesn't fix your issue, I recommend performing a Windows XP system restore.
Also, make sure your wireless keyboard is not disabled or has any errors with it in device manager. A red x next to that keyboard would indicate it is disabled. You can re-enable it by right clicking it and select "Enable" from a small menu.
If there is a yellow exclamation point next to that wireless keyboard then that indicates a problem with it.