I would not risk hanging a glass shelf with those strips. They are meant for hanging things directly from (like a towel, hair dryer, etc.), not a shelf that sticks out. Instead of just pulling down on the strip, it will pull down and away, almost a guaranteed recipe for broken glass.
Unfortunately there is no way to anchor something to the wall without proper anchors. If you are able to get the shelf anchored into a stud then the hole should not be all that large - it's the drywall anchors that tend to make larger holes.
If the wall was concrete you might be able to glue it to the wall, but this will leave a larger mess (leftover glue that someone has to scrape off) than screws in drywall.
Perhaps you could offer to repair any damage you are responsible for? Otherwise you should get a free-standing shelving unit.
Sounds like your sheetrock is somehow damaged from moisture or the EZ Toggle would have certainly worked. You now need to install wood backing, or move your speakers so the brackets can be screwed into the studs.
To install wood backing, first you need to cut out a rectangle of your wall's sheetrock slightly larger than the intended size of the backing panel, with the vertical cuts centered on studs. Don't try to save the old piece, replace it with new. I would use plywood instead of sawn lumber which would make a cold spot. Cut the plywood to fit flush between the studs, preferably with face grain horizontal.
Cut 2 narrow plywood strips to act as cleats. Using construction adhesive, nail/glue the strips to the studs set back from the front edge equal to the plywood thickness. Once the glue has set, nail/glue the plywood backing to the cleats. You are not supposed to nail into the edge of plywood, but in this case it's just to hold the panel in place until the glue sets.
Install a new sheetrock piece to fill the hole you cut out. Tape and mud the joints. Prime paint your finished mud job. If you have textured walls, and don't have a mud gun, there are spray cans of wall texture available for repairs just like this. Either way, you'll need to practice application to match the existing texture. Once dry, apply paint top coat to match existing wall. Whew!
Sure you can't just move the speakers to the studs? ;)
Best Answer
I agree with @user263983. Use plastic drywall anchors that go with the size of flathead screw that fits properly into receivers on the backs of the speakers. The heads of the screws must be small enough to fit inside the bottoms of the slots and large enough to be retained in the smaller tops of the slots.
Drill square to the wall so the screws are perpendicular to the wall. Drive the screws in so that the bracket just barely slip fits between the underside of the screw head and the plastic anchor or the wall.
If you don't have a drill or don't want to drill, there are anchors with a sharp end that you force into the drywall and then twist with a screwdriver to pull into the wall. Get the right size, and use two for each speaker. Self drilling drywall anchors