Hey, Mia. I'm not sure why I think the following applies in a way, but maybe it makes sense.
A long time ago, my school has some serious, I mean serious, morale problems, due in no small part to the awful administrator we had at the time.
A few of us who had some training in Systems thinking tried to make some headway with Peter Senge's philosophy's about learning organizations....(he wrote "the Fifth Discipline" and has some wonderful things to say about organizations). Anyway, we took everyone through a process of identifying where we were as an organization...it was pretty hard stuff as emotions were running high.
As we went through this.....what we found out (and is pretty well outlined with this type of thinking) is that its simply not enough to identify where you are as an organization (and this is where somehow, I think it applies to relationships) or what the problems may be..........there must be a common goal that is actively WORKED ON...not just talked about.
So when you said you asked your brother to spend time with the kids, it hit me in the same way...there must be an action taken to move forward.
In the case of our school, it was our common focus of being a postive environment for children.....with the WORK or movement forward after that identification of a goal....of getting on with really DOING IT.
Ok, long way around: you support your brother as you love him dearly. You have made that clear to him. Now the actual demonstration of that comes with: you working for him when he needs you, him taking your kids to the park, etc.. It's the love in ACTION that moves us beyond just discussing it.
Anyway, sorry for the ramble, but it seems you are a good sister and whatever your brother's issues are....demonstrating that love in action is going to move him and you closer to a goal (showing your love and support...him finding healing in some way?) than just being concerned.
He is a lucky guy, I think, to have a sister like you.