I was reading about Tay Sachs disease - it is essentially the deficiency of the enzyme hexosaminidase. Why can't the disease be treated by administering the enzyme (prepared artificially/extracted from other living things)to the patient?
In general, yes, administration of enzymes in some form or another can be used to treat a deficiency. A common example of this is the use of lactase to treat lactose intolerance. Since lactose intolerance results from the body not producing enough (or any) lactase, you can take oral lactase to alleviate the symptoms:
Lactase products are tablets or drops that contain lactase, the enzyme that breaks down lactose. You can take lactase tablets before you eat or drink milk products. You can also add lactase drops to milk before you drink it. The lactase breaks down the lactose in foods and drinks, lowering your chances of having lactose intolerance symptoms. - NIDDK
For some deficiencies, a general strategy used is gene therapy, which provides a functional copy of a gene to an affected person's cells. The advantage of gene therapy is that, rather than repeatedly dosing with enzyme, you give the cells the ability to make the enzyme/protein that is lacking, which results in a longer-lasting therapy. This can also facilitate delivery to hard-to-reach places (e.g. you might not want to repeatedly inject enzyme into the liver, or brain, but if there's a vector that can target that tissue you can just inject it intravenously).
I'm not overly familiar with Tay-Sachs, but there is some good information in the comments that one of the commenters will hopefully expand into an answer.