USB ports on HP Ultraslim docking station do not work

Remove the laptop, turn off the docking station.

Now, plug in the keyboard/mouse etc

Now turn on the docking station, give it a little time, and now put on your machine. It appears many docking stations need to work in a certain order, it doesn't detect the keyboard/mouse (I guess in the same way a PC doesn't detect a PS2 mouse/keyboard without a reboot).

Some have reported that, "disable one of the two USB Root Hub from the Device Manager and refresh the list allowing the system to recognize the connected devices". Source

Others have claimed to roll back the driver to a generic driver Source

Based upon your comments, since it doesn't recognize multiple devices (keyboards/external disks) then it sounds faulty. You could ask HP for advice directly, if it's in warranty then it's a no brainer (send it back) but you may need to get a new dock :( However, to test this, try a different laptop in the dock (if possible).