How can I stay longer on emergency leave with my brother who has terminal cancer?

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So first, what your going through is neither easy nor fair.
I Left my duty station in Germany to be with my father as he died but was convinced (rightly so) to not burn up all my leave so I still had some available after he passed (an unknown like your brother, week, month, 4 months) I spent 14 days with him then went back to my unit...45 days later I went on leave again to take care of his afairs....

So that said... If you have significant time remaining on your current contract.

Your commander can grant you advance leave, ut not more than you can earn in your remaining enlistment.
"4–4. Advance leave
b. No more advance leave may be granted than the amount of leave the Soldier will accrue during his/her remaining period of active duty. "

Next, no commander I know would approve 60 days of advance leave even if they could..for a variety of good reasons. Though you can ask, all they can say is no.

No one but HRC can transfer you to a new duty station and that's not going to happen super quickly, and would need to be done back at your unit in any case. Understand it is just a request, though you have one of the qualifying reasons to ask so have that going for you. While you are still there, speak to his doctors and get the attending physician’s signed statement providing the specific
medical diagnosis and prognosis, including the date of onset, past and anticipated periods of
hospitalization, periods of convalescence, and anticipated life expectancy for terminal cases. You will need that if you decide to request Compassionate Reassignment

You would submit an application, DA Form 3739, Application for Compassionate
Actions through your chain of command, CO, BN, BDE, DIV, and then on from there to HRC. The governing regulation is Army Regulation 614-200, Chapter 5, Section III. https://armypubs.army.mil/epubs/DR_pubs/DR_a/pdf/web/ARN14314_AR614-200_FINAL.pdf

There are only two people that can answer this question: Your 1SG and Commander. After 50 days already...they may recall you back to base. You need to be prepared for that command.

SGM Erik Marquez gave probably one of the most thorough, accurate and informative answers you are going to find.

The only thing I can see as a possibility which he did not discuss is a TDY. In order for that to work, you would need to have a local unit who had work for you, AND you would need your current unit to agree to it. And regardless, you would have to go back to your current unit for at least a couple of days to sign in, do paperwork, and sign back out. I couldn't even begin to tell you how to make it happen, but if you were able to find a local unit, AND if you could convince your Commander (I would guess BN level, but maybe BDE?), I am pretty sure this is something that would have to be coordinated by most of the staff - S1, S3, and S4 would all be involved. Which is a lot of extra work for any almost definitely already overworked staff. Chances of this happening are next to zero - but not zero. In the words of Lloyd Christmas, "so you're telling me there's a chance." As SGM Marquez said, the worst they can say is "no."

I will just add that you have already been gone for 50 days. With no definitive end state in sight, expect to head back to your unit. The primary purpose of the emergency leave in these situations is to give you a chance to say goodbye. This may sound harsh, and it is not intended to be, but if you haven't managed to say goodbye in 50 days, then another 50 won't be any help. I know you want to be there until the end, but you have already been given more emergency leave time with your brother than I had seen be granted to anyone in my 20 years of service.