Based on the model name and K-code (yours is a K2) I will guess it has a Dana/Spicer transmission. While not too complex, there are a number of parts on the trans. that could have failed or worn out to cause the problem. But first, confirm the shifter is fully moving the change arm on the transmission case. It needs to fully move in order to shift from 1-2 and back from 2-1 again. Clean any dirt, debris, etc. from the change arm and inspect the cable. A stretched, worn or mis-adjusted cable can also limit the ability to change gears. If you are convinced everything outside the trans. is all good, there may be an internal fault with the transmission.
Unfortunately, there are no service parts available for this older model...only the entire transmission as a unit (includes rear axle) and it lists for $336.21. For that kind of cost, a gently used mower would be a better value than to keep this one going.