Jesus' words (v.9) that he's "not asking on behalf of the world", but only "those whom you gave me because they are yours" don't mean he doesn't care about 'the world', but are expanded a few verses later, in the present passage into his prayer that the world will be changed by those he is praying for. Jesus is remarkably focussed, even under the pressure of the moment.
We, those of us who believe & belong, have a huge responsibility to Jesus for those who at this point don't. Also, as well as any appropriate social action this responsibility may involve, we have a responsiblity to keep our pretty wide-ranging 'prayer lists' focussed. And re-focus our list from time to time in keeping with any responsibility we discern Jesus giving us for others.
On another tack, I suggest that Jesus is not praying for 'Christian Unity' here. Not firstly, that is. Thanks to the arrogance & intransigence of some of us, & our denominations, 'Unity' is a bit of a joke, that we rationalise by making a foray from time to time when we're embarrassed into so doing by taking part in some ecumenical venture or another. No wonder 'outsiders' don't take us seriously when they see us not taking each other seriously, not taking Our Lord seriously. It's because Jesus sees the joke's on us that he prays as he does. He's the victim of that disunity of faith among his own people that will bring them down even more than it brings him down. The oneness Jesus prays for here is far more than some lip service to ecumenism!
The issue goes back to The Beginning. To the first principle of Creation (not the fundamentalist kind!) that everything & every- one is created good. A consequence of humans being made in God's own image is that that image, including its oneness with God (& therefore with each other) is to flow on to benefit the whole Creation. We cannot be one with anyone else, not even our own self, unless & until we are restored to the oneness that has always been God's plan for Creation. That only happens as we are restored to grace by the New Adam (who also has an Eve side!) Mostly our efforts at 'church unity' go nowhere because we don't make Jesus' connection of the centrality of oneness of Being, before any attempted oneness of doing.
That we have a privileged view of Jesus himself at prayer here is our
opportunity to focus on the Oneness his praying is all about - becoming
as one with God as he is. As one with our fellows as he is. One 'bit' that's
struck me more than before is where Jesus prays (in v.24) that "those whom
you have given me may be with me where I am, to see my glory, which you
have given me.....before the foundation of the world".( The Creation connection!)
What now strikes me is that Jesus is praying that we (those God has given
him) will be with him where his glory is, i.e. wherever he sacrifices himself.
Not back then, in our case, but now. Jesus prays we'll stay the distance
& show our love for others in their situation as he shows his for us
all in his, even though his kind of glory is painful & costly!