This may not be the correct solution, but if you have another compatible power adapter available (friend; order one and return, if not), try that. I had no idea it could happen until it happened to me, but sometimes the adapter fails in a way where there is still power getting through or maybe it's just battery 'juice' (in your case the blinking light), but not ENOUGH to power up the system. I used to think power adapters were either full-on or dead, but in my case, my laptop would power up for only a second or two then turn off. I thought it had overheated and fried, but after a couple of days I decided to test with my husband's adapter and BINGO. Ordered my own and good as new.
Anyway, may not be THE answer, but if readily available, it doesn't hurt to test it out.
Disconnect the power cable and then remove the battery from the laptop. Leave both disconnected from the laptop for at least a minute. Then, put the battery back in the laptop, connect the power cable, and try turning on the laptop.