Some examples of D-Rings, or strap hangers. Far more secure than screw eyes. All of us who restore antique frames have seen damage from screw eyes failing. Screw holes should be drilled for the strongest installation. If wire is to be used, the hanger should optimally be angled in the direction the wire will go. On larger frames, two hangers matched to two wall hardware, is best. This requires some precision in measurement and installation. Securely wrapped wire loops can be attached to the D, allowing some adjustment to the wall screw or bolt. For really heavy objects, I usually make cleats, sometimes known as "French Cleats" from plywood. Two strips of ply with matching angles cut on the edge, attached to the frame and the wall so they can hook together.
Now, if you insist on screw eyes, at least wire them right.
1. Through the eye
2. Around the shank, and below the wire.
3. Back through the eye.
4. Wrap the wire around it self.
This makes the wire as low as possible, reducing the force against the one little screw. Physics. Also, as the route the wire follows is more complex, this makes the wire less likely to pull free.
There are commercially available other forms of frame hardware, some that include mechanisms for leveling.
Apparantly, mechanical competence is a hindrence to artistic genius, grumbles the old curmudgeon.