Catch the cat if you have any place where you can keep it for a few days. Advertize in the paper - yes this costs too but some local papers will allow advertizements for free if the person advertizing is not charging. You could also advertize online at your local
freecycle group.
Say something to the effect.
"Friendly outdoor cat, excellent mouser. Would be good addition to a farm or somewhere in the country. To give away."
Then get rid of the cat yourself.
Otherwise I liked the idea (sounded easier) of putting the cat on the humane society's doorstep.
I'm surprised that the humane society did not at least give you any other options for getting rid of this pest. You could call the police the next time your screen is damaged. If you explain to them that the animal control people refuse to help and that this is a dispute between you and a neighbor then they have to intervene and at least speak to the woman. She's is inadvertently causing damage to your house and property and refuses to compensate you for it - that could be construed as vandalizm. It is in the interest of law enforcement to at least investigate this and mediate for fear that an altercation could develope between the two of you.