You don't have a leak. If you did, your CO2 would all be gone (gauge pressure down ~0) after 6 days.
Couple of questions:
- Did you refrigerate the keg?
- Did you purge the headspace of air, and if so, what was your process?
It takes more pressure to carbonate to a given level at room temp vs. cold. There are some fairly simple formulas, as well as tables and on-line calculators, that will tell you how much carbonation you will get at equilibrium for any combination of pressure and temperature. If you didn't refrigerate, your carbonation level will be lower than expected.
The gauge pressure needed to achieve any given carbonation level assumes that the headspace is 100% CO2. If you don't purge the headspace of air, then the actual CO2 pressure (rigorously the partial pressure) will only be about 1/2 of what is required for a gauge pressure of 15 psi. So, not purging the headspace of air will result in lower carbonation than expected.
The video linked recommends three purge cycles, which will remove ~88% of the air if purging at 15 psi. This is adequate to get reasonably close to the expected carbonation level, but not enough to protect the beverage from oxidation, if that is a concern (I don't know how important that is for kombucha.)
Carbonating to 1 "volume" of carbonation requires about 2 g of CO2 per liter of beverage, so 1 gal requires about 7.6 g of CO2 to get 1 volume. Typical carbonation levels for beer are about 2.5 volumes, so carbonating to that level requires about 19 g of CO2. Pushing 1 gal of beer at 15 psi will require about another 2 volumes, or about 15 g of CO2.
Now, you may have some residual CO2 in the kombucha from fermentation, but the actual level will depend on whether the fermentation was open or closed (i.e. with an airlock.) If open the residual CO2 will be close to 0, but if closed should be around 0.8 - 0.9 volumes, which will reduce the amount of CO2 required to carbonate to the desired level.
A 16 g cartridge is about enough to carbonate and push 1/2 gal of beverage, but 1 gal will require 2 cartridges.
Brew on