When should I use a ground plane?
A lot depends on whether you are doing a commercial PCB to be manufactured and sold, or a few one-off boards for your own use.  If the former, then a  ground plan is more important to keep radiated emissions down to meet FCC (Part 15, unintentional radiators), CE and other regulations, although I've seen plenty of commercial products with one or two-sided boards and no-ground planes.  (On the other hand, one of my clients just made a six-layer board with three signal layers, a Vcc layer, and two ground planes in a sandwich configuration.)

If you only have signals up to 400 kHz, then it sounds like you aren't using a microcontroller (or if you are, you are using an internal oscillator so there is no crystal).  If you have prototyped your circuit on a wireless breadboard or the like, and it works okay, then it should work fine without a ground plane.

Even if you are doing a commercial product, if your device is battery powered,
and it has no signals above 1.705 MHz (which is true in your case), then it is exempt from FCC rules.