If you want a hardware-only solution to duplicate what you have, a search on
keyboard wedge hardware serial
brings up some products that will make serial data look like data from a keyboard. Some also convert to USB, saving the expense of a serial/USB converter.
The Silicon Labs kit looks like it enumerates as a generic HID, not a keyboard, so you would need to provide an application to read the data.
For the DB9 - PS/2 - USB adapters idea to work, I think the DB9 would need to receive serial keyboard data, not raw serial data.