This situation happens in an ICU somewhere every single day, but unless it involves a celebrity, we usually don't hear about it. The fact is that in the case of brain death, most hospitals will give the family a certain amount of time to understand and absorb the reality of their loved ones' injury. I think they do respect the family's desire to hold out hope-for awhile. If the patient is brain dead, and is declared so according to very specific and well established criteria, they will usually give the family a designated "reasonable" period during which the family can speak with doctors, clergy, counselors, other family members, etc. After this period has passed, they are legally within their rights to withdraw life support, and they do. It only occasionally happens that families challenge this legally, and when they do, they usually lose.
Now if she is NOT brain dead, hospitals do not impose this decision on the family. In the absence of legal directives, the next of kin has the final say. At that point, yes, the family has a right to keep hoping, at least to the degree that they have insurance or the means to pay for treatment. That happens every day as well.
We at WS do not know whether or not BK actually meets the criteria for brain death. If she does not, she will probably be moved to a rehab/nursing facility, potentially for many years, because it does not sound like her family is leaning towards discontinuing life support as long as they retain their capacity to hope.