Did it ever work better?
Often the thermostats on portable heaters are not very precise because they are so heavily influenced by the heat from the heater rather than the actual temperature of the air in the general area. There's not a whole lot that can really be done about it short of having the temperature sensor located rather farther from the heat source. It may help a good bit to have a fan blowing on the heater to circulate the air around it more.
Since the heater is cycling on and off in response to the thermostat setting, I rather doubt it's damaged.
It would not be an impossibly difficult project to put together a remote thermostat cord setup. You could either wire it through a residential thermostat for electric heating directly, or use a battery-powered digital residential thermostat hooked up with a solid state (or I suppose mechanical) relay and possibly some appropriate power source for the relay control line. The latter setup would allow you to use a small, low-voltage and low power wire between the thermostat and the rest of the setup; a telephone wire, for instance, would work fine and be readily available in various lengths.