Is it that entire brass threaded bit? (Admittedly, I do not know sodastream anatomy.) Is the brass bit simply screwed in too tight?
If it just needs to be unscrewed, I would use a trick similar to removing header studs on an engine, but modified.
Find two nuts that are the same size thread, but completely round off the corners on one of the nuts. Screw on the unmodified nut half way. Screw on the modified nut as tight as you can against the first nut. Use a socket wrench to apply torque on the first nut and it should give you just enough friction to unscrew the whole thing.
Its hard to tell via pics but that brass thing is pretty big. Like, bigger than a standard screw. Needle might work but hard to get leverage on anything that deep into it - tho that might be inexperience.
Is it that entire brass threaded bit? (Admittedly, I do not know sodastream anatomy.) Is the brass bit simply screwed in too tight?
If it just needs to be unscrewed, I would use a trick similar to removing header studs on an engine, but modified.
Find two nuts that are the same size thread, but completely round off the corners on one of the nuts. Screw on the unmodified nut half way. Screw on the modified nut as tight as you can against the first nut. Use a socket wrench to apply torque on the first nut and it should give you just enough friction to unscrew the whole thing.
Needle nose pliers might work too.
Its hard to tell via pics but that brass thing is pretty big. Like, bigger than a standard screw. Needle might work but hard to get leverage on anything that deep into it - tho that might be inexperience.
Even if you can get on it with some right angle long needlenose you risk tearing up the brass.
Then there is heat. Using an oxy acetylene torch to heat up the whole assembly in an attempt to loosen the threads might work. What could go wrong?