[robespierre]

In that picture of the old TV set, the square box unit looks like a voltage tripler. That is a Cockroft-Walton multiplier made of three stages of cross-connected diodes and capacitors.
You will find that—especially on older large sets—where the flyback transformer output isn't high enough. It might also do HSTAT and X-ray protection if that is a Trinitron tube, where the HSTAT electrode is coaxial around the second anode socket, and the red anode cable is also coaxial.

There's a lot to learn about CRTs, so you're in for a great learning experience!
Let us know how you get on, as I can share some ideas, and hopefully other readers as well.

Just a note about cleaning the tube itself: the dark gray color on the bell is a water-based carbon paint called the aquadag. You don't want to clean that too vigorously or use any solvent or it might come off. Just a light wipe with a barely damp cloth is all that's needed. The metal band that surrounds the face of the tube should never be removed, because it provides the integral implosion protection. When you disassemble the bezel, be careful about supporting the tube because you don't want it to fall on its neck, the most fragile part. Keeping it face down on a towel is the best plan.

I can see that you were able to remove the anode cap, which is difficult to do without directions, so you've already shown great aptitude at CRT repair.

An apparently failed flyback can be a case of cracked solder joints, since the flyback is heavy and puts a lot of strain on the circuit board.

See if you get any ideas from these guides:
http://www.repairfaq.org/sam/crtfaq.htm
http://www.repairfaq.org/sam/monfaq.htm