Normally you put a magnetic material between them. It becomes a magnetostatics problem instead of an electrostatics.
This core of magnetic material directs all of their magnetic field to be shared, minus a small leakage. You need this core for high coupling coefficient. What you want is called a transformer.
Here is an image of what you need:
I did debate about this answer being a comment so don't bust a gut voting down (or up) but I had too much to say to fit a comment.
Can you get 3A from those batteries? For now I'll assume you can...
Assuming that you are producing no-more than 80V on the primary (because the FET will break if you are doing a 100v) and given your turns ratio of 1:160, and given the "imperfect coupling factor" (that I pull out of the experienced side of my brain) I'd say you should produce about 5kV.
The FET is getting warm and this isn't a worry but it does rather hint at your 12V supply rail falling to somewhat less when it is powered on. Maybe you can measure it. My suspicion is that it might be falling to about 6V or 8V and this will not efficiently turn the fet on and this could be causing the FET to warm. If this is true then maybe you might only be producing 1kV.
Of course, there is a strong possibility that you damaged your FET with over voltage the very forst time you used it. I note on your earlier threads/questions you were advised to use a 200V device with lower on-resistance.
I'll also point out that a lot of these circuits are from unprofessional sites that might have (with luck) got something to work and easily misconstrued their own circuit diagram thus misleading everyone who tries to build one. Also notable about the website that contains the circuit is the lack of information about the transformer and how to wind it.
The original design came from: -
http://www.geocities.com/CapeCanaveral/Lab/5322/fbt2.htm
But don't bother trying the link because geocities is no more.
Looking at your pictures and knowing how difficult it is to get a step-up transformer working with ratios over 100 I have to point out that there is no hope for the way you have it connected. 555 needs to drive the transistor which provides voltage to the FET and the FET source, emitter of the transistor and 0V of the 555 have to be "rigidly connected". The currents you might be inducing in 0V connections could easily mean you get less than 200V out of this design.
. I converted 310V DC to 12V DC as you can see.
I wrote spice model for T1(Transformer)
Best Answer
You've made the beginner's error of not understanding the importance of dot-notation when implementing a flyback transformer design. In your first circuit you have, in effect, got lucky with the right output voltage but it's not acting as a flyback converter but more like a regular forward converter and using the transformer as a regular step-down device.
This is not how flyback designs work.
The lower picture shows how I've altered the secondary coil to suit it working as a flyback converter. Flyback converters work by charging the primary coil and reverse biasing the secondary diode then, turning off the primary MOSFET so that "fly-back" occurs and the energy stored in the transformer's magnetic field is released and forward biases the diode. It's a two-phase operation.
Correct use of transformer secondary phase relative to primary phase: -
Notice the change in the position of the dot on the secondary - you can keep the dot as per your original diagram but you then need to connect the secondary diode to the non-dotted end of the secondary (as shown in my amendment to your original diagram).
Picture from A Guide to Flyback Transformers by CoilCraft.
Other reading: Mean Well - Flyback converter: -