How to unclog a toilet? (solid object obstruction)

I would first try to replace the flapper, it may be old and worn and not creating a proper seal. If that does not work try replacing the ballcock (float valve), as these too can become old and worn and not function properly.

You probably (and hopefully) have a large amount of sand/grout stuck in the P trap. If so your best bet would be to remove the P trap and give it a good thorough cleaning (out in the yard, with a hose).
(While the trap is off you'll be able to inspect the tailpiece and the drain line as it goes into the wall and clean those up too if necessary.)

If you can't detach your P trap you have a few options:

• If your trap has a cleanout nut, open it up and run water through until gunk stops coming out (you can employ your wire hanger here too, to "encourage" it).
  (My experience is most older pipes that have such a cleanout nut are so old that the thing is frozen and can't be opened without breaking the pipe, so be careful if you try this)
• Remove as much of the tailpiece as you can and clean out the trap with a shop-vac and/or spoon/wire/etc.
• Work a snake down the drain and whip it around like crazy while running LOTS of water
  (In the hope that you'll wash enough of the grout further downstream that (a) your drain runs quickly again, and (b) it doesn't clog your pipes further down the line where you can't get at it.
  This is really a last resort and just delays (or moves) the need for a proper fix...)