Because of our negligence towards the issue since so long, ...
Am I using this correctly? I want to convey something like 1-2 months ago or since around September.
When used in a "time-related" context (as opposed to when it means because), the word since is invariably followed by a reference to an actual time (or event) in the past. For example,...
You might just about get away with...
...but even adding ago wouldn't make OP's example acceptable to me (though it might for some).
As noted in various comments, the normal usage here would be for so long.
Exactly how long ago so long might be is totally subjective. In some contexts, you might use it to refer back to a time less than an hour ago. In other contexts, people might say that several months (or even years, decades, or centuries) wasn't actually so long ago.
One can say “since long ago” understandably and idiomatically, or “for so long” as mentioned in comments, but “since so long” is too awkward for words. As FumbleFingers noted in a comment, “since so long ago” is rare in Google Books. Looking at another corpus – stackoverflow with its approximately 14 million questions, and several times that many answers and comments – “since so long” shows up only 9 times (as of Dec. 2012). Instead, the term of art on SO is the curious “since long time”, which appears about 1150 times. For example, in questions #9590114 line 4, #12569673 line 11, and #13727862 line 1, we see