The bank robber had me lie down on the floor.
Der Bankräuber befahl mir, mich auf den Boden hinzulegen.
The examiner had me recite a poem.
Der Prüfer ließ mich ein Gedicht rezitieren.
Only people can "have you do something": it means that someone in authority or a position of power tells you to do something.
"Agony had me suffer" would mean that agony was a person who ordered you to suffer.
"Die Liebe ließ mich leiden", by contrast, would normally be translated as "Love made me suffer" or "Love caused me to suffer". Not "Love had me suffer", unless you are being poetic, and saying that love is like a powerful person who ordered you to suffer.
What is it you wanted to say, johyleIN?