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? ;)
No those anchors will NOT be sufficient to hold up THAT shelf. Those shelf brackets are too short and there will be a LARGE pulling force on the upper screw.
Assuming from the image that's a 10 inch shelf and the screws at 2 inches apart, hanging 79lbs on the edge of the shelf will get you a whopping
395 LBS of pull force on the top screw.
When you look at that picture and look at the metal end, does it remind you of any tool that is sitting on top of your toolbox? A claw-hammer perhaps?
You may get away with the type of anchor that mushrooms out behind the wall as shown below. Remembering to buy the size that matches your drywall thickness. But I do NOT recommend these either for that shelf.
With the shortness of that bracket, I'd still be concerned that was not enough to hold up much weight. If those fail, it is going to pull down a rather large piece of drywall too.
Personally I would be inclined to add a backer board out of a nice piece of wood, the wider the better, using one of the following two methods.
Option 1: Outboard screws... Board cut to fit to next stud width. In your case 3 (Don't forget to add a 2x4 width.). Mount board to studs then centre shelf on the wood.
Option 2.: Inboard Screws. Cut board to about 1 inch wider than the shelf. Then centre the board on the studs as shown in the image below.
Which method is best?
Use the method that gets the screws in the wood closest to the screws in the brackets. However, be aware that with the "inboard screws" method, the end of the board will tend to pull away from the wall if the screws are too far away from the brackets.
If the boards spans more than two studs, I'd add another couple of screws at each stud.
If you, or the Mrs, don't like the visible screws, use one of the many capping or plugging methods.
Best Answer
The standard advice is to use the stud finder, mount the bike hooks on studs, and make do with a few lost inches. But if you must get those extra inches, you can mount a 2x4 on the wall horizontally, with at least each end anchored into a stud. And then you can install your bike hook on the 2x4.