OK, here we go <takes deep breath>:
First off, instead of connecting together lines 9 to 10 and 13 to 14 (mode 1+3), you want to connect lines 11 to 12 and 13 to 14 (mode 2+3). This should get you a centimeter further (progress is progress, nonetheless).
Secondly, One thing that I have noticed after I reviewed the entire topic is that you stated above that after you built your adapter, you could only get an image (although green), on one of at least 2 tested monitors. This tells me 2 things.
1. The monitor that you were able to get the green image on actually supports sync on green, and that your LM1881 chip is not necessary with this monitor.
2. It is possible that your LM1881 circuit is still not built correctly. The fact that you got no image at all on the other monitor means that it does not support a Sync on Green signal (and is possibly at this point, exactly what your xbox is still outputting) and thus, your LM1881 is either not connected correctly, was damaged during this project, or was Dead On Arrival.
Note: There is a very slim possibility that this monitor (the one that cannot get a signal at all) is just very picky in that is does not like the non-standard VGA signal that the LM1881 produces AND ALSO does not support S.o.G. This leaves very small (but existent) doubt to second point (2), however I am 85% sure that point 2 is true.