#1 LED is ALWAYS on when power is connected, and indicates that the power supply is at least connected.
#2 LED is (I think) when the power button is pressed and the system begins to power up. The LED indicates that the power supply is activated, and the logic board has received operating power.
Unfortunately the LED continues to blink, so I believe that means that the power supply is not activating.
First choice for the repair is replacing the power supply. I think it is a likely fix.
If not, then it probably gets expensive.