It can't be done. That is my guess. I think you would need a pressure suit or cabin.
I am a former USAF weather officer with flight experience. I have a Masters Degree in Meteorology and a BS in Space Physics. I have working experience in analyzing the jet stream and also using data for vertical airspeed (or omega, as some call it). That does not give me magic powers or even make me generally correct at any time, but I think it provides me with a rough familiarity to review the question and mention the key issues it raises. I ask for your trust in my meteorology and flight experience when I say that you should not assume you have good reason to trust in your idea just because some of the answers got a lot of thumbs up or seem like they good answers. Based on my training and experience, none of the answers that predate mine have justified your idea. The point of that is to keep you skeptical, not to conduct ad hominem attacks to support my own answer. I don't care if I get one up vote.
I think with a pressurized suit of some sort and a massive soaring aircraft, some day a circumnavigation will be completed. It's almost inevitable, but how and when is the question.
I think your idea won't work unless you use dynamic soaring. Period. You seem to be under the impression a trip might succeed using the vertical velocity of the atmosphere for lift. However, across broad areas where lift would be needed, the atmosphere would rise at only a few cm/s. For scale analysis, let's pick a number and say 3 cm/s. As the air would be rising and bringing your craft with it, your kite would only maintain altitude with a descent rate not exceeding 3 cm/s. (In reality, due to other issues, probably MUCH less than 3 cm/s would be required!). How achievable is a 3 cm/s descent rate? Well, let's look at a glide slope ratio analysis and compare the result to existing soaring aircraft.
If your kite is extremely efficient, you will probably not have a very fast forward air speed. After all, the aircraft's drag goes up with the square of velocity (assuming constant drag coefficient) so you would probably travel with low air speed. Let's say you are traveling forward at just 60 knots, which is close to 30 m/s. That is also 3000 cm/s forward air speed. If you had a descent rate of 3 cm/s, your ratio of forward speed to vertical speed would be 3000 cm/s to 3 cm/s, or 1000:1. How does that glide ratio compare to world-class sailplanes? Well, an extremely efficient open-class glider might have a glide ratio of 70:1, so your design would have to get FOURTEEN times more distance out of the glide slope of the best aircraft. Put another way, you are trying to be FOURTEEN TIMES MORE EFFICIENT at utilizing energy. That may be possible, but you are talking about a massive change in design technology.
All of this scale analysis assumes you are always in an area of robust uplift. Unfortunately, the atmosphere is not like that, not even on the south side of "THE" jet stream. In fact, where the jet stream speeds up, those sections are called "jet streaks" and they have descending air in two of the four quadrants that surround them (and likewise for jet sinks). So, it's not some easy picture as a simplified schematic suggests. Your aircraft would constantly go through places of descending air as well. So, it seems to me that your aircraft would need tens of times better glide slope than an efficient glider. That's not a very easy path to success.
Instead of using rising air, I think you would need to copy the technique of countless seabirds who use dynamic soaring. They are able to travel much farther and more quickly than birds that only depend on updrafts. It's a simple analysis, but nature seems to be pointing the way!
Dynamic soaring around the globe would require a nearly continuous trail of highly differential shear. Where the continuity is broken, you would be losing altitude until you found another location of differential shear. Above the tropopause, perhaps that is possible with a sufficiently advanced aircraft. I doubt it is possible below the tropopause, where areas of no lift and occasional turbulence would both be enemies of your kite. So, I leave a scale analysis for dynamic soaring possibilities to someone else. I suspect it would be a more likely path to success, but still it would require massive improvement in soaring efficiency as well as highly detailed knowledge of where the differential shear is located.
Also, I think you would definitely need pressurization or else you would need a lot of specialized breathing training and still take on a lot of risk to your life and limb.
Without cabin or full suit pressurization, you would need to breathe pressurized oxygen, which means forcibly breathing out against HIGH positive pressure on EVERY breath for days on end. Your mask would be crushed against your face. Goodness knows how your eyeballs and other body parts would eventually react. I have done some of the positive pressure breathing in a USAF pressure chamber (Peterson AFB). It didn't seem sustainable. I have seen videos of this positive pressure breathing in actual aircrew exercises. It looks quite challenging! And that is only at altitudes that might not be high enough for your kite idea to succeed.
Breathing out against this pressure for long periods of time causes health problems and is exhausting. I even wonder if such a long period of positive-pressure breathing could kill you through embolism or some other form of respiratory-related injury. Even with sufficiemt positive pressure to drive up the partial pressure of oxygen in your blood, if you got too far above the tropopause, no amount of positive pressure would even keep you alive. So, your flight would be limited to areas below 40,000 feet, which means you could not circumnavigate close to the equator where the tropopause exceeds that altitude.
As for your general understanding of a physical path to gliding 'round the globe, "THE" jet stream is ALWAYS just a term for RELATIVE velocity distribution. What looks like a continuous jet stream around the world might be a stream with 300 knots over Alaska but then only 60 knots over the Atlantic! That's why looking at numbers and not just colors and lines is important. Likewise, the horizontal and vertical gradients of the "horizontal" velocity would vary widely, as would vertical velocity. I can definitively say that in all my of days doing manual analysis of the jet stream (before computers were left to do it alone), I found it unusual to find a continuous jet across the Western Northern hemisphere, let alone fully around the globe in either the Northern OR Southern Hemisphere. Now imagine looking for a jet stream AND significant differential shear along the jet. It would be quite an endeavor. It may be possible, but the idea would require in-depth study of the wind patterns, not just for theoretical feasibility but likelihood it would be feasible to find necessary and sustained conditions on the special day an aircraft would be ready for launch.
The idea of circumnavigation by natural power alone is an old one. However, there is always room for more achievement. I think someday someone might circumnavigate the globe in a special gliding craft, but I do think they will use a pressure suit or cabin.