Sometimes you can get the screen to flash, briefly, when pressing and holding the on key.
This can happen when the tablet battery is pretty much completely depleted.
It should boot when plugged in to a charger, but some (most?) chargers are too wussy to supply all of the current the tablet wants. So the tablet starts up, decides the charger ain't cutting it, and shuts down again. (Or maybe it's the power supply system of the tablet at fault - it can't draw more than a certain amount of current. Or both.)
One solution is to use a USB battery pack instead of a wall charger. Battery packs are rarely limited in their output current, and can deliver 2.5A or more if the tablet demands it. This can sometimes let the tablet boot normally and it will start the normal recharging process - and besides which, it will recharge faster. Banggood often has Xiomi packs on for good prices, I recommend everyone to pick one up.
Another solution may be to use a high-current USB charger. Be aware, however, that most "2A" or "2.5A" rated wall chargers are nothing of the sort. Most will barely source 500 mA or so, meaning they're no better than a "standard" charger, and may still fail. Also, the Cube and most other tablets can't draw 2A anyway.
Finally, you can try the rubber band trick noted in the comments in this YouTube video. Basically, plug it in, then hold down the on key for 5-20 minutes. This is obviously best done with elastic bands. This seems to force the tablet to stay active, or at least enough to jumpstart the charging process. The author says 20 minutes but I've had success with only 5 minutes of holding.
(I know I saw this elastic band trick mentioned elsewhere, but I can't find it again. Comments on the alternative/original location welcome.)
Of course, you might find - as I have recently - that the deeply-depleted pack of the tablet IS in fact charging - just very slowly. This is typically done for safety purposes, to avoid battery overheating and other issues. Holding the power button doesn't seem to affect this, at least not always. So leave it overnight and see what happens.