nebbian said...From memory most switch mode inverters and power supplies run at around 90% efficiency -- so inverting and then converting it back isn't all that bad.
ymmv though

Yeah, I agree. The switchmode in the monitor is not going to care much about the type of inverter so you can use a cheap one. Using the closest wattage to the monitors requirements are going to make it the most efficient. It also makes it easier to change to a new/old LCD when the first one cops out eventually.
For the laptop itself, there are a few converters that go from a nominal 12v to whatever the laptop needs.
To answer your question on whether you can connect to 12v internally in a monitor, it is unlikely that they use a 12v rail and then regulate that to other voltages. They would just go from "220v" to whatever different power rails are needed, and there will be at least a few. Probably a 12v, 5v, and a 3.3v, plus whatever the backlight uses.
The cost of a cheap inverter is going to be cheaper/easier than your time to understand what is needed, if it is even possible.
The few monitors I have seen with external plugpacks go to something like 18v. Don't ask me why, but it might be one of the main supply rails needed.