It sounds like she already had an aggression problem and at 3 years old it may be something you can't change. It is certainly something you don't want to 'practice' on with a 10 year old cat.
One thing you can (and should) do is discourage any live animal chasing, including squirrels. Offer her rawhides, toys and rope.
You need (really, you need to do this) have her evaluated by a professional who can decide whether or not she's got a chance at all of living in a multi-animal household.
Your local humane society has people typically on site who evaluate dogs that come in there as rescues. Give them a call and ask for help with yours.
Many local vets should also know of truly qualified 'behaviorists' in your area. Be aware that 'anyone' can call themselves a behaviorist and many do! It's vital to check references and look for people who come recommended by professionals in the medical and established rescue fields.
Finally, PetCo and Pet Smart have dog behavior classes offered (usually by private parties who work out of the stores). They may be valuable resources in deciding whether or not your new companion can be calmed/trained.
If she can't - it might be the better option to find her a place where she really will be the only animal and not exposed to children, cats or anything else that she may consider weak enough to chase.
Good luck with her. I commend you for trying so hard and being so persistent! You are truly HER best friend!
This link may have some helpful hints too http://www.seefido.com/html/untitled37.htm
Many have forgotten this truth, but you must not forget it.
You remain responsible, forever, for what you have tamed.