Sure it can be done, but one question, is this something that you actually want to do, or is it a hypothetical question (e.g a homework question)?
The first part of getting a circuit to stay on after a brief pulse is easy if one uses a relay. Wire the momentary N.O. switch to turn on the relay, and wire a N.O. contact from the relay in parallel with the momentary switch. Then when the switch is pressed the relay will operate and the relay contact will keep it on even when the momentary switch is released.
Now the problem is how to get it to release in 5 to 10 seconds. For a real application I'd recommend an electronic timer, but if you want to stick with passive components I can think of a couple of options off hand. One would be to put a a fairly large vlaue capacitor in series with the relay coil (which would have to be fiarly high resistance) while the capacitor is charging the relay will be activated, but once it is charged the current will fall and the relay will deactivate. The problem here is getting a 5 to 10 second delay, 5 to 10 milliseconds would be easy (though that is a bit short for a mecanical relay) but the longer value could require a rather large capacitor.
Another posibility is to use what is sometimes called a current relay (sometimes used for starting certian motors) , and put its coil in serires with the nicrome wire, as the nicrome heats its resitance increases and the current falls, and if if you could get just the right compnent values the relay could drop out once the nicrome is hot, but getting just the right values might be a problem.