No display

May be failure of RAM. Please check it first.

First, plug in a external monitor to confirm the condition of the machine, after if not display appers, your laptop is damage in post (only in HP to recovery) or HDD die (replace a HDD).
If the displays on in external monitor,your graphic card is damage (leave in HP ou replace a Video Card).

This could be lack of power - recharge or replace battery or run direct from mains.
Or it could be Hard drive failure.
If it is HDD failure you should have got P.O.S.T showing so I go for lack of power.
This looks like a service centre job

Check your power cord/PSU and all connections between the laptop and the wall!

This may also be a sleep/hibernate problem, from ACPI control. The "other" blinking LED makes me suspect that as the MOST likely cause!

1 To test this theory, press any key hat looks like "sleep" or "suspend". Wait 20 seconds to see if laptop wakes up.

2 if not, then press the power button and HOLD IT DOWN continuously for 10 seconds. If the blinking light goes off, we have discovered the problem! If so, continue with step 3.

3 Now, the computer will be "off" entirely, and we can do a "cold start". Press power button again, and wait for start up as usual. Note if the HD light has come on now, (and flashes to indicate the boot process is working, if not, stop and let me know this -or replace HD with a new/known good one to proceed)

4. When/if the laptop reaches normal desktop, check the battery level FIRST. If it is NOT full (assuming the plug has been in all this time, battery SHOULD be charged) your battery may be damaged. Go to step 5. Otherwise, if battery is good, go to step 6.

5 Turn the laptop off normally, and let let it charge for a a few hours, to determine this. Start and check levels. If it is not showing greater charge level (and does not fully charge after some more time) then the battery may be bad. Check voltage with VTVM, or if you do not know what that is, take battery to a repair shop and ask them to do it. Voltage should match or exceed the levels stated on battery labels.

6. Regardless of the cause, be aware of this fact: The greatest friend a battery has is ACPI/Power Saving "sleep/suspend/hibernate and related HD spin down settings). But it can ALSO be a battery's greatest enemy in this situation:

NEVER disconnect the power cord for more than a few seconds when you laptop is suspended/sleeping or hibernating! In fact, it is best to relocate the laptop when it is awake, to be sure removing the power is registered in BIOS. Why? While the laptop uses less energy in these suspended power states, if the laptop is left without AC power (bad cord, bad transformer/PSU or bad connection in wall socket) for an extended period while it is asleep, the battery has no "shut down" ability, so it depletes the battery PAST its safe depletion levels. If the laptop is "awake" when this level is reached, it shuts down automatically. But the paradox is, most laptops must BE awake to do this!