ssh config file
Make your life easier - reduce typing by putting your ssh options into a configuration file.
Login to to your Linux computer located inside Engineering's network
Create the following file and call it .ssh/config
Host gate
Hostname gate.eng.cam.ac.uk
User crsid
Host your_work_computer_name
User crsid
ProxyJump gate
For Windows, use ProxyCommand instead of ProxyJump:
ProxyCommand ssh.exe -a -x gate nc %h %p
Once that works, you can copy your public key using ssh-copy-id gate and ssh-copy-id computername.
If you are on Windows 10, you don't have the ssh-copy-id script, so open .ssh/id_rsa.pub in Notepad and copy the contents to the ~/.ssh/authorized_keys on gate and on computername. Be careful not to split the long line into multiple lines.
Using Microsoft Remote Desktop
Create the following file and call it .ssh/config
Host gate
Hostname gate.eng.cam.ac.uk
User crsid
LocalForward 3391 computername:3389
Then open the Remote desktop on localhost:3391.