I have an Nikon 18-105mm VR lens. Is it make sense to buy a 18-200mm VR II lens?

I already have an 18-105mm lens for my D5200.IS it make sense to buy an extra 18-200mm telephoto zoom les? Which one is better 18-200mm or 70-300mm?

In your case, the 70 - 300mm will be better.

Yes, sell the 85-105mm and buy the 18-200mm. Not only will it give you more reach, but the optics are better.

Alternatives to selling the 18-105mm is to keep it as a backup, sell/loan it to someone who got the 18-55mm lens.

I own all three of these lenses.

The 18-105mm lens is being used by my eldest son with my D90.

I have a 18-200mm lens that I use for vacations, but only for vacations. It is not sufficient for everyday use as like all super-zoom lenses, it is not as "optically pure" as a less powerful zoom. However, I have found that if I do not use this lens at apertures wider than f/8, it will produce acceptable results.

As a result, it is a daylight only lens.

The 18-200mm is NOT optically better than the 18-105mm. Having both lenses, I can tell you this for sure. The main advantage of the 18-200mm is that it has a metal mounting ring, where the 18-105mm has a plastic mounting ring.

The 70-300mm lens is a optically better than the 18-200mm, and has a bit longer reach. However, there's not as much difference between 200mm and 300mm as most people think. There's a noticeable difference, but not really significant.

You can compare here:

http://www.althephoto.com/concepts/lenses.php

My advice; if you are happy with the 18-105mm and want a telephoto lens - and are not opposed to occasionally changing out lenses - get the Nikon AF-S 70-300mm (do not get the Nikon AF 70-300mm version, it is not a very good lens - and will not autofocus if you have an entry level camera).

But if you want an all-in-one lens, especially for vacation where you do not want to carry a lot of gear, get the 18-200.

The part of your intended 18 to 200mm from 18 to 105 would be 'wasted' since it is already covered by the 18 to 105. Get a tele zoom instead, maybe 70 to 300mm.

70-300 will suit you much better.
I use 18-135 and an old 100-300 most of the time on my Sony.

If you get the 70-300mm, be sure to get the GOOD AF-S VR model, not the cheapo. Besides the issue of autofocusing, the 24MP D5200 is going to magnify every optical defect.
The 70-300mm is going to make your camera front heavy. It helps if your tripod has a good sturdy pan/tilt head like the Manfrotto 804RC2 or 460MG 3D. The latter has an advantage that the pivot point is about two inches forward of the tripod socket. One little disadvantage of the 460MG 3D that I have found is that because of thermal contraction the azimuth adjustment gets balky in subzero weather.