The little box in the kitchen will be your controller.
When your heating / water system is working normally, are you able to turn on a hot tap and get hot water straight away?
If so, I suspect you have a combi boiler, and suspect it is in the loft.
Have a look at your fuse box / consumer unit. The boiler may have its own circuit (unlikely, but worth checking), and maye the fuse has blown or the circuit breaker has tripped.
When your 2 year old is asleep, can you have a look at your loft hatch. Does the loft hatch have a pole thing to open it, and release a loft ladder for you to climb up?
If you can get in the loft, look at the boiler and see if it has any error messages, or maybe the fuse in the plug connected to the boiler has just blown and needs replacing.
If there is a friend or neighbour who could turn on a hot tap while you look at the boiler, listen and watch it to see if it fires up when the hot tap is run.
Do post back with any other info you can, and between us we might be able to suss out what's wrong.