Probably because at every stage where a court has looked into the issue they have crushed him. The UK Supreme Court's ruling was very clear on the matter and did him no favours in terms of public opinion. Taking to the ECHR and losing would be another serious blow.
I think that's the correct answer. So, rather than going up the legal hierarchy in Europe, Assange is attempting a political end-run by going to the UN.
The bigger question is why is the UN going along with it.
They already have basically no credibility - why make it worse?
The UN should stop trying to tell countries what to do, no one listens to them anyway. Focus on helping when asked, and brokering voluntary treaties, when asked.
