Why has Electronic Arts been doing so poorly lately?           You'd think that the giant of games would be able to squash the competition

IDK
Probably bad management and/or they're buying too many other studios.
They made tons of very good selling games, but they still have enormous high expenses.

EA has spent a lot of effort on bad investments into questionable new IPs...

I applaud them for trying, but they really need to do more upfront research to make the right investments.

No mega hits except for Madden and Fifa and those licences must be incredibly expensive.

They also bought a lot of assets whose value must have plummeted over the past couple of years.

Because Saboteur, Dante's inferno and Dead Space Extraction flopped?

Out of these the only real flop where they lost quite a bit money was probably The Saboteur.

FIFA 10 alone made enough profit to make up for all those flops.

Yeah, you would have thought the ridiculously high sales of FIFA 10, which I think is probably on track to become the best selling FIFA ever, should have given them a decent amount of profit, even with the greater investment in the game

Dante's Inferno's ad campaign was huge.

You gotta remember a lot of these games they have to pay enormous fees for licenses. Exclusivity for FIFA, NFL and Beatles have got to cost an arm and a leg. Just because these games sell a lot doesn't mean EA is the one making the profit.

Because they have a lot of games that sell a million or so, but not many that sell many many millions of copies. The ones that do are on so many formats and thus have large porting costs and have expensive licenses. Plus they buy a lot of studios.