Led light turns on for a while and goes off

Where is the LED< you never said. (exactly)
monitor or PC, ?
you posted a laptop to a forum called monitors
which is it, a laptop , desktop or only a monitor.?
toss coin, land LT. 2700 (legacy relic with square screen floppy and windows 98 first ones)
take out the battery , the big one, it must run forever on AC power pack or its a hardware problem. (there can be 2 main batteries)
1 bottom and on the right rear, CD out, and battery is its place.
yes, that is what they did then if you have it. I cant see yours.
Next do the HP long reset.
all batteries, out, now. (bottom big, right rear, out)
power pack out. disconnected.,
push the power button from 10 to 30 seconds, this is the HP classic, long reset in every manual they make and on there service page, 1st thing.

next is power pack, if still dead. the goal is getting the F10 key
BIOS to work first. it must work at all times at power on
power button on , hammer f10. and see bios
if fact it told you to press F10 for 16 years now, every day.
2001 laptops have zero value. now. zero. upgrade to 2008 is better, put this one in he trash. go to goodwill store upgrade for $20? (they wont want yours, sorry, its too old)

so do that. is bet but if maniacal. ? H3LL bent to spend $120 or more with a HDD bad, on a $1 PC, here goes.
OS + HDD = $170
if the screen is dead now say so, say totally black screen
what it did before it died, is not important unless the fan is dead is it? (all PC's ran ok before now)
then ask for help,
windows will shut done only if it is corrupted. (self)
hardware has many cause of shut down, about 10 causes.
one is the PC overheated from not cleaning it every year.
the run down. BSOD (black)
big battery out, AC pack only, got that?
this PC has a place for 2 main batteries both out now.

The PC has CMOS cOin cell, not one ever lasts 16 years.
so that be like the first thing id replace (basic service this is)
open bottom hatch cover to see it.

1: fan dead or all vent and to it blocked in lint as all are this old.
1b: fan dead not packed?
see slide show of nasty LT here.
http://www.pcdied.com/overheated.html
2: coin cell dead, above 2.9v ok, below never. remove it replace
3: power on hammer f10 the screen most show , "hit f10 to enter setup" if not bad screen or far worse (yes 3 billion transistors inside can fail, main board, or shorted modules inside up to 8)
4: now the flash light trick this old dog has CCFL back lamps.
do this.
http://www.pcdied.com/monitors.html#flashlighttest

if that shows DATA, then the CCFL are bad.
if no data now, the problem is more serious

http://pcdied.com/black-screen.html#Laptop

if it still is dead, we replace the power pack
or remove all modules in the PC that can short killing power dead.
any thing inside can short. 4 ways in fact, DC power, data, address or control buses. to find that we pull things, to find it.
ask. im not teaching Fault isolation checks here. but that last link above has it, see key word stripping there (control+F)

if all you wanted was to guess, say so, say no diagnosis guessing.
for that I say it needs to be cleaned of a pound of dust inside and new coin cell and dump the 2 useless main batteries, if both there.

left is power (both), right is battery failures.
for the coin cell, (its hard to get to on this PC)
read page 63 (pdf) here
the service manual. is here, for free, do not pay scalpers for it.
https://www.manualslib.com/download/273070/Hp-Compaq-2710p.html