I suspect it depends what kind of cheese the whey came from. Whey has all of the water soluble components of the milk. It loses the casein and fat. How much of the albumin and lactose it loses depends on the cheese. If the milk was heated enough (190F I believe) then the albumin will denature and not be in the whey. If the cheese was acidified with a bacterial inoculation then the bacteria will have converted some of the lactose into lactic acid which will both be in the whey. Riboflavin is water soluble and is what gives the whey that greenish tint.
Either way- most of the nutrients that bacteria like are not in the whey and it is always highly acidic. I have personally kept whey for a couple weeks with no ill effect.
I usually use my whey soon by making ricotta. If the cheese it came from didn't denature the albumin in the milk then you can gently heat the whey until the the albumin precipitates out. A gallon of whole milk makes almost a pound of mozzarella and then a little less than a cup of ricotta. That's a lasagna right there.
In general, you shouldn't trust anything with a poor seal to last outside the fridge. In the fridge, they'll last just as long as a well-sealed jar that you've just opened. Commercially processed pickles can last a year in the fridge. I wouldn't expect homemade ones to last as long but they should still be good for long enough to eat them all, at least months!
The solution in the future is simply to switch to a different canning method, for example this lower temperature one or a boiling water bath one. Here's an example recipe - you could certainly adjust the spices to match the recipe you like, but it's of course safest to stick to that trusted recipe for the bulk ingredients.
Your problem is that this canning method doesn't reliably produce good seals. The similar inversion method, mentioned by Martin in the comments (turning the jar over after filling and closing) does work better, but even that doesn't always make a good seal! See for example this USDA source, which says:
Some other methods of sealing jars call for inverting a closed, filled jar of hot product for anywhere from thirty seconds to one hour. (Inverting is turning the filled jar upside down on its lid.) While this inversion process can be successful in producing a sealed jar, it works best with very hot product. Individual variation in practicing this procedure or unexpected interruptions can result in delays between filling jars, getting lids screwed on, and inverting the jars. If the product cools down too much, the temperature of the product can become low enough to no longer be effective in sealing jars or preventing spoilage.
When the inversion process does work, the vacuum seals of filled jars still tend to be
weaker than those produced by a short boiling water canning process. A larger amount of
retained oxygen in the headspace may allow some mold growth if airborne molds contaminated
the surface of the product as the jar was filled and closed. More complete removal of oxygen
from the headspace also offers some longer protection from undesirable color and flavor changes with some types of fruit products. A weak seal may be more likely to fail during storage.
Best Answer
As the brine is the preserving agent (due to the acidity, even in a sweet pickle), not having the cucumber in the jar will not reduce its shelf life.
The brine should easily last six months (and probably much more) in refrigerator.