The PS3 and the Wii U, two consoles which faced a lot of struggles after their launch. We know that the:

-PS3 launched after the highly successful PS2 causing Sony to become arrogant and lose their way. The Wii U also came after Wii's exceptional sales.

-PS3 launched at a ridiculously high price. A lot of people believe Wii U is overpriced still now.

-PS3 launched with not a lot number of major first party titles. Wii U also had the same issue of not having enough must have games at launch from Nintendo.

-PS3 received one crap 3rd party port after another, most of which ran better on their competitor's console, leading a lot of people to believe the 360 was the better choice to play 3rd party games. Nintendo faced similar issues with 3rd parties where they got lesser ports in the beginning and eventually those dried up as well.

-PS3's architecture was very difficult for developers to program for. Wii U also has an architecture that is different from its competitors, making it take more time and effort for devs to port their games to it.

Now the issues the PS3 and Wii U faced were not necessarily identical. However, we know that both consoles struggled. Even then, despite all of the issues the PS3 had, and we know there were many, Sony still somehow managed to turn the PS3 around, making it sell over 80M lifetime and massive install base. They even set the PS4 up for success by garnering the goodwill and trust of gamers that bought the PS3. The Wii U on the other hand isn't recovering at all, and doesn't look like it ever will, probably going down as Nintendo's worst selling home console.

Why is this? What did Sony do to turn the PS3 around that Nintendo isn't doing with the Wii U? If so, why isn't Nintendo doing or trying the same to right the Wii U ship?

Because Nintendo took too long to turn it around.

3rd party

They didn't want to.

Really though. Sony took a big financial hit to right the PS3 ship. I'm not sure they made that money back during the PS3 era, although you could argue it help set them up for success with the PS4.

Nintendo is a fiscally conservative company, and they're pretty smart. They were able to make money during the Gamecube era, and they're back to profit now. If it's a choice between growing install base and making profit, they'll generally take the profit.

Simple. Playstation brand > Nintendo brand in the console department. 2 out of 3 console sold over 100m and ps4 is on pace to be the third.  Even at $500/$600 ps3 was selling at minimum 80k per week.  I don't remember the lowest week.

Sony kept third party support on their console and also hardware sales weren't too bad for the PS3, they were pretty neck-in-neck with the 360.

Price is one of the biggest issues. The biggest problem with the PS3 was the initial price. Once they dropped the price people wanted to buy it. As shown by the Wii U's sales, not many people do want one. It doesn't matter if it's the cheapest option. When it's the lowest costing console you can't just keep price dropping something that no one really wants.

The PS3 wasn't selling nearly as bad as the Wii U. The PS3 had the chance for very big pricecuts (which made it desirable), the Wii U doesn't have room for many/drastic pricecuts (and it was never popular). Wii U had droughts in it's 1st 2 years, PS3 didn't ever (at least that I know). 3rd party support was always with PS3, Wii U was lucky to get 3rd party games (now it isn't even getting most). And, of course, the japanese market has shrunk a lot.

https://www.youtube.com/watch?v=7ARtdqYB4_g#t=9m40s

this guy explains it very well

By throwing away their money.
Nintendo tries to be profitable, Sony not so much...