We purchased new blinds. We are having difficulty installing the brackets.

The screws won't hold in the wallboard, even with the smallest of anchors.
I have pre-drilled holes. There is some kind of metal underneath, which the metal drill bit won't penetrate.
If it were a stud, imagine the drill would go through the wood.

There doesn't seem to be much room behind the drywall. Two other larger plastic anchors both split when I screwed them in for the opposite bracket.

Interesting, the small metal anchors we used for the other set of blinds worked well, but not for this bracket on these blinds.

Trying to remove a screw that got stuck in the anchor, it pulled out all the anchors, and tore the drywall.
When I filled the drywall with the 3M Patch Plus Primer, this time I set the 4 anchors inside the spackling and let it dry for nearly 24 hours. The space is about an inch square.

I used some smaller screws to hold the bracket in the anchor. When another screw got stuck in the anchor, and wouldn't go farther in, I tried to remove it and try another screw. Gently pulling it out, all the anchors came out again and tore the patch, which doesn't seem adequate to for the repair.

I could place more patch or more joint compound on it, this time adding the anchors after the spackling/ joint compound dries. The spackling application fixed one of the holes on the other bracket. I pre-drilled a small hole and screwed-in the small anchor. Am not sure this will work on this side.

Am getting frustrated trying to fix this. It may take some revision of the wallboard, something I don't think I'm capable of doing. It may be worth calling in a handyman/woman.

I want to get your advice on what I should do to properly anchor the brackets.

I am including some photos of the anchor problem in the wallboard, the bracket on the opposite side of the window as well as photos of the patch and anchor. To show the curve of the window opening, am including a photo of the installed blinds in another room, which has a similar window shape.

 

Those cheap short screws and plastic anchors are useless for this one.
First thing I do is throw the cheap plastic anchors in the trash.
Just use longer screws.
No way is does that "patch kit have any holding power!
100% sure you have the drill turning in the right direction, even a cheap steel but should be able to drill through sheet rock and wood.
If for some reason the window is framed in metal using a titanium coated or cobalt drill should go right through it.

 

I am not sure what the metal you are hitting is but use a small drill to get through it and then larger drills to get it to the size you need.

This works when you are trying to get through the side wall of a steel stud where the side wall wants to bend.
Be careful as it might be a protection plate for wiring. This is highly unlikely but stranger things have happened.

Also I have never used 4 screws, just two on the diagonal have done the job.
Put all 4 in using anchors and you are just weakening the drywall.

 

You have 4 holes in the back side of the bracket. If that's solid, use them instead. Otherwise you might need to just use longer screws (2 1/2") on the sides, not anchors. Or pull the brackets forward an inch or so and try again. Or fasten a board across the window mullion first, then screw into that if there are only steel studs on the sides. Those anchors aren't going to hold in spackle, sorry!