The Kawai K1II has three MIDI ports-- MIDI THRU, MIDI OUT, and MIDI IN. To use the K1II as a sound module, you'll need to connect a MIDI cable to the K1II's MIDI IN port, because you want it to receive MIDI data coming IN from the PX-350M. See page 9 of the Owner's Manual for the K1II, in the section on "Sample Setups," and look at "(a) Playing the K1II from a digital piano."
The Casio PX-350M has two MIDI ports-- MIDI OUT and MIDI IN. To use the PX-350M as a keyboard controller, you'll need to connect the other end of the MIDI cable to the PX-350M's MIDI OUT port, because you want to send the MIDI data coming OUT of it to the K1II.
See page E-45 of the User's Guide for the PX-350M, and note the sections on "Keyboard Channel" and "Local Control." Although the PX-350M lets you split or layer tones together, it apparently transmits only one keyboard channel, so you might as well set the Keyboard Channel to 01. To use the PX-350M as a keyboard controller, you'll want to turn the Local Control to Off, which will prevent the PX-350M from making any sounds of its own when you play on its keys. (Just remember to turn the Local Control back on when you do want to play the PX-350M's sounds!)
The settings on the K1II look more complicated, and since I'm not familiar with the K1II I can't give you detailed setup advice. But you'll want to study section "3. MIDI Receive Parameters" that begins on page 42. Be sure to set the RCV CH to the same channel number which you set the PX-350M to transmit on. And if the PX-350M is sending on only one channel then I assume you can turn the OMNI parameter to OFF.
As for the PGM parameter, it might be best to set it to OFF so the K1II will ignore Program Change messages coming from the PX-350M, since there's no correlation between the tones on the PX-350M and the patches on the K1II. That means you would need to select the desired patch on the K1II.