Charging light blinking but not turning on

[Do2log]

I have HP EliteBook 8440p that I have been using for a while on the charger only (without the battery in). Recently, when I plug the AC charger in, the charging light blinks (orange light and sometimes blue light) but the laptop is not turning on but after many trials of plugging and unplugging the charger, it finally turned on (this happened many times).

I tried an advice to spray some alcohol in the charging port (in case it needs cleaning) and it worked for a while. But now I have to try many many times till it eventually turns on.

N.B:

• When it turns on, it's working completely fine.
• When it turns on and I can use it for a while,  after turning the laptop off:  
  If I unplugged the charger (either from the wall or the laptop itself), the same thing happens the next time I turn it on (charging light blinking and not turning on). But if I leave it plugged in (both the electricity port and the laptop), it works fine the next day.
• I have tried: Bios update and reset, disconnecting charger and battery and hold the power button for 30 seconds, connecting the battery (wasn't charging then), trying another charger
• I got a battery warning few months ago that the battery needs replacement but I didn't replace it yet as I was working on the charger only with the battery off.

Thanks a lot and I hope someone can help me!

[Do2log]

Any suggestions??