[LunarBerry]

To get a yorkie costs a good bit of money, to keep it healthy and get everything it needs, is also a good bit of cost. By wanting to get a yorkie, you'd have to be able to do more than just home-care, it costs lots of money. It's like having a child/baby in the closest, literal sense.
Not saying you couldn't be responsible "parents" but by wanting a yorkie, you have to take financial responsibility and/or show an incredible amount of care. Either way, someone would have to bare the financial costs and/or bare the care of the pup. There's no guarantee the pup may grow up completely healthy, even with all the care it can be given.
It sounds like you have done some research, which is good. But understand that if it comes down to the yorkie having a health issue (again they can develop, even from good breeding and so on) you have to be prepare or willing to handle that. I wish you the best of luck as I can actually understand your circumstance. Either way you need to discuss this further with your mother and work out a solution for whatever is needed. You would have to show her you are responsible and having the ability to pay for it's care is a plus. Your mother may have her own reasons, but as long as you live in her house, you must have her agreement or abide her wishes.