One way to get very low currents at higher voltages is to do something like this:
But as you notice, that particular circuit generates a lot more than you are asking for. But the main ideas are these:
- Chop up the DC into AC so that it can be multiplied up using a Cockroft-Walton topology.
- Apply the Cockroft-Walton multiplier.
In this case, they also use a transformer to help get much higher voltages. But you don't need to. Let's just use your circuit, slightly modified, to get something similar:
simulate this circuit – Schematic created using CircuitLab
Please note that I didn't add enough stages to get anywhere near the voltage you want to generate. I suspect that you might get close to the voltage you want with something like 24 to 30 capacitors and diodes and even then you'd only get perhaps hundreds of nanoamps. (Should be relatively safe, though.)
Here is an example of what a nicely layed-out (for high voltages) Cockroft-Walton might look like:
Now, you could also go for the transformer model. But then you need to worry about some decent design of the transformer; or, at least, some consideration of the design details if you use an existing transformer (an audio frequency transformer is often used because common oscillation frequencies will be in the audio range.)
You could also consider the Joule Thief design. This is where I think it may be possible to do things from a junk box (or used-up CFL light bulb.) But I'd need time to think more about that and then to test out the ideas. There are a number of factors to get right in trying to do more than just light an LED using such a design. But you can look up the basic construction ideas on the web.