The ThinkJet is VERY simple. The ThinkJet printers (and clones) do not employ any type of printhead covers or 'priming stations', so the cartridges are prone to drying out if not used for a while. A quick 'priming' is usually required, even after only a week or two of non-use.
The cartridges also tend to leak if placed in odd positions or subjected to rapid temperature changes. Make sure the cartridge has not drooled on itself and caused ink to cake down on the contacts in the holder. Clean the gold contacts GENTLY with a cotton swab moistened with rubbing alcohol.
BTW: Like motor oil, fresh ink is great for cleaning up old, dried-up ink.
The ink is contained in a rubber bladder inside the plastic shell. There is a hole in the 'butt-end' of the cartridge. GENTLY stick a bent paper clip in and push on the bladder to prime it. A drop of ink should form on the printhead. Use a piece of tissue to wipe the drop off and re-install the cartridge.
Note that there are two types of HP ThinkJet printheads. One is the older, original type meant for printing on special 'ThinkJet' paper, and the newer 'Plain Paper' ones meant for, well, plain paper! ('PLAIN' will be printed on the side of the cartridge).
The older cartridges printing will appear very light if printed on plain paper. Make sure you have the 'Plain' type. Note that even with a 'plain' paper cartridge, the printing is lighter than a Laserjet or DeskJet, especially in draft or single pass modes. Don't expect razor sharp printouts. This was the first Inkjet printer!
As for the missing jets, eyeball the cartridge contacts and see if they appear straight and aligned correctly. The contact area could have slipped and may be out of alignment (although rare).
The flex cable/connector assembly is held in place with a pair of plastic bars. The "bars" have two pins that snap into the 'carriage' (they also provide alignment).
If the contacts appear to be out of alignment, carefully pull out the plastic bars to release the contact pad, realign the holes and press them back into position. Make sure the rubber 'bumps' behind the contacts are clean and undamaged.
(This makes more sense when you actually see it :^))
Be careful! Nothing needs to be forced.