Finding the optimal voltage for a sealed transformer
It's probably a switching supply built to drive a neon tube (they require a high output impedance since the tube itself has a negative resistance characteristic). The device would rectify the 230VAC input to create about 300VDC and then chop that to get 10kV using a high-frequency ferrite-core transformer. I would expect if it worked you might get about 1/20 the output with 12V in, which is what you are seeing. 

If you search for "neon power supply 10kV" (without the quotes) you'll find examples of this kind of product. 

So you should be able to feed it about 30-35VDC and get 1kV out, but it will be deliberately poorly regulated for load changes (more than a handful of mA drawn will significantly drop the output voltage). 

That's just an educated guess.. it's hard to tell what's inside there.