No, Nintendo isn't what it once was. But it's stupid to ask "do they still exist?".
Their second-to-last console, the Wii, which was replaced in 2012, is the third-best selling console of all time behind the first two PlayStations. Yes, it was an underpowered fad with a low attach rate, but it was still a success.
Their second-to-last handheld, the DS, which was replaced in 2011, is the best selling handheld of all time, even almost outselling the PlayStation 2. Yes, they've never had a serious competitor in that market, and smartphones have pretty much killed off said market at this point, but it was still a success.
Yes, their latest console, the Wii U, did almost everything wrong. For comparison, in its last two years (2010-2012), the Wii sold more (almost 15 million) than the Wii U has during its entire lifespan (over 13 million). But 13 million isn't an absolute failure. It managed it without ever getting a significant price cut, and had a high attach rate.
In addition, their latest handheld, the 3DS, overcame a bad launch to end up selling almost as much as the original NES.
And outside of hardware, a franchise that Nintendo owns 1/3rd of, Pokémon, is experiencing its second wave of mass popularity after its 90's debut.
They currently have nearly 5 billion dollars cash-on-hand, negligible debt, and their stock is 3 times higher than its lowest point. And as for future endeavours, the Switch reveal video has over 22 million views, over 500,000 likes, and the Jimmy Fallon video is nearing 5 million views. We will see if this initial positivity can continue when the launch lineup and price are revealed in January.