They released the DS because the PSP was coming. Plain and simple. 

The GBA's 2003 was monstrous, 20+ million in shipments, higher than any year for the 3DS. 

The NES and Wii also took advantage of windows of oppurtunity where th competetion wasn't great. Apple/Android wasn't around when the Wii was launched, nor were Sony/MS even remotely offering anything for that audience. Nintendo had *years* alone with that concept until about 2009 when Kinect finally came out and the iOS shop started to really open up (Angry Birds would launch in December 2009). 

The NES forced third parties to make games exclusively for Nintendo through the lock-out chip which effectively froze Sega out of the market. I remember the NES was my first machine, but back then I didn't know much about Nintendo or Sega and my friend was telling to make sure to get the Nintendo because it had the far wider variety of games (racing games, hockey games, basketball games, adventure games, etc. what he was really saying was it had all the 3rd party support, even though neither of us knew what that meant back then). 

When Nintendo is confronted with direct competetion they tend not to do too well, Apple/Android is directly taking away casuals and kids from Nintendo. 

The Genesis and then ultimately the Playstation/XBox took away the kids who grew into teengers and then adults from the NES days, aside from a loyal remaining fanbase. 

Nintendo can't compete for soccer moms/girls, they're not sitting around waiting for Nintendo. A smartphone is 1000x more appealling to them then a Nintendo product, it fits their social lifestyle much moreso. They can't match a smartphone's functionality -- does a 3DS Twitter? Take really nice photos? Instagram? Text message? To be honest from what I gather outside of Mario, women tend to prefer iOS games too, they don't want Metroid or Fire Emblem or Zelda ... that's too heavy for a casual player. Candy Crush, Angry Birds, Farmville, etc. are far better at getting the job done. The iPhone is a fashion accessory, the 3DS and Wii/Wii U are toys by comparision. 

Are you saying it's an opinion if one number is bigger than another?  No this is when we use the term "Wrong."  MarioKart 8 sold 2.65m and Donky Kong: TF hasn't even sold 0.7m.  DK is NOT a big game.  Fact.

If we're looking at the Wii U, nothing without Mario in it is a "big game". And even certain Mario centric IP are having a hard time -- Mario & Sonic was a big seller on the Wii ... a big dissapointment on Wii U. 

Let's look at the name of the thread shall we...

Why would Nintendo make another console after so many failed attempts?

In no way is this question relavent to sales, or facts. It's an opinion based thread based on the failure of the Virtual Boy and the Wii U (maybe) vs the success of the NES, SNES, n64, Gamecube, Wii, GB, GBA, DS, and 3DS. 

In this case you are wrong, numbers are not a measure of what is considered big. It's how critically acclaimed those titles are on the consoles in question. So if you want to argue with me on Opinion vs Fact I will gladly take you up on that. In the end, opinion isn't wrong, but you are wrong for using facts in a thread that started with someone's opinion. I provided my opinion, against someone elses opinion and if you don't like it you can kindly leave the thread. Simple eh?

Also, Mario Kart 8, including digital sold over 3 million so your facts are wrong. Try again. 

Sorry I laughed myself sick on that one.   Yep Destiny is a "Small" game for Activision with its $1,000,000,000  in revenue within the first week since its review scores weren't perfect.  You are the definition of someone who lives in their own world.

Or I live in a world where people have opinions and aren't afraid to express them. Destiny was caused by hype, end of story. Also Activision did not make 1 trillion dollars in 1 week from Destiny. Not only is that impossible that would honestly requires the entire gaming industry to buy the the game 10-100 times over. You're just picking apart my statements because you have no opinion of your own. 

This isn't true. The losses from the gamecube and n64 were hidden from the massive profits of their handhelds.

https://news.ycombinator.com/item?id=2481975

Did you read your link? It's using the a vague statement about the 3rd year of the N64's life. At this point, there wasn't much for them to lose money on as there weren't many big projects later on in its life. 

Once again, all the numbers they're quoting are towards the end of each console's life and include the extremely successful handhelds. Sony's gaming division lost all of its money on R&D and the launch of the PS3, they were able to make money later after they were able to drive down the costs of production. You can believe that their consoles were held up by their handhelds, but I don't see any reason to (and I surely hope you don't think this only includes the literal cost of production subtracted from the retail price).