I have used one of those air con kits, and they are not of much use if you are having the sort of problems you describe.
First of all, the compressor will NOT operate at all if there is insufficient gas in the system - it has a pressure sensor switch which prevents the clutch engaging. So, don't bother trying to bridging low pressure valves etc.
You have no alternative other than to go to a garage (or mobile chappie) with an aircon machine. A gas charge will cost between £50 and £75, depending on where you live. (Down South here, it's towards the upper end of that price scale). The machine will first discharge the system and create a vacuum. I understand that some machines give a warning if they cannot achieve this due to a system leak. My personal experience is that the warning comes when the machine puts in the gas and the system pressure cannot be maintained. Before you request the charge, make sure that they add dye (should be standard if they know what they're doing). This will show up leaks quite clearly undera UV light source.
You can probably rule out a compressor fault - they normally suffer failed bearings and then with a lot of noise. Discarding that, then, the first and foremost source of your problems is probably a gas leakage. It could be anywhere: the condensor, the receiver/dryer or even, as I have had, a leaking service valve (which does not show up when the machine is charging the system).
You should be aware, though, that AC systems contain both gas and oil (for lubrication). The aircon should be activated at least once a month but ideally should be used pretty much all the time. If it is not used (as yours hasn't) the seals in the system tend to "dry out", causing further leakage faults.
Fingers crossed it's just a minor leak that can be fixed with a 60p "O" ring. For sure, though, you'll never find out until you get it properly gassed.
Let us know how you get on.