If there is a problem with any syringe, they'll usually give you a new one for free. I've never had a problem getting a replacement. Call the pharmacist who dispensed the prescription and ask for advice.
It's EXTREMELY unlikely you'll hit an artery - they're too deep for the short needle to reach, under layers of fat and muscle.
Can your Mother not do the syringe herself? Personally, if you're injecting someone else i would go for the middle of the upper arm. Pinch the skin between your thumb and index finger next to it, and put the syringe in at a 45degree angle. That insure that you don't have to put force on the needle to get it in, it'll glide under the skin. There's nothing to worry about, you can see veins through the skin, and arteries are way to deep. They make the needle length for a reason.. It's to short to hit anything major. If you do get blood in the syringe, it's possibly because you've hit a blood vessel. This isn't a big deal!! :) Just carry on injecting and apply pressure to the injection sight once you've removed the syringe. You'll be fine :) Have confidence in yourself!
If your Mother can injection herself, or even if she can use one hand.. there's a new mechanism out called the "MyClicPen" It's a pre-filled syringe in a pen form. Almost like a diabetic needle? You only need one hand to perform it. It's simple. You take off the cap off, press the end of it onto your leg (for example) it'll move down as the pen is worked on a spring. then you press the button and it'll fire the needle out and automatically inject you. It takes seconds to do and is really effect for someone who has hand mobility problems. The only problem i found is that i thought it was a bit rough.. it literally fired the needle out! causing me to jump and bruise and little afterwards but someone who is used to injecting into their leg (i inject into my arm or stomach) would have tougher skin, so may not bruise and find it easier. I hope this helps!