Install a VNC server on the desktop. Install a VNC client (sometimes called a viewer) in the phone. Connect the phone to the desktop by wifi (you'll need the IP address of the desktop - hit Windows key+r, type cmd and Enter, and you'll be in a command window - type ipconfig /all and you'll see your IP addresses - for Ethernet if you're connected to the office network that way and for wifi).
The phone will be able to control the desktop (the desktop will also be able to control itself - so the keyboard and mouse are still active), but if you use handwriting to create text when you're in a document on the desktop (and you're using the phone), it should put the text into the document. That is, if you can use handwriting-to-text on any app, not just on a select few. If it only works on the phone for a few apps, the app may have to be written to accept input from that handwriting-to-text app, and it won't work with any other apps or any programs on the desktop.
Google Handwriting Input may work in that case.
