Why did the League of Nations fail?

The United States never became a member. America was the most powerful country with the largest economy.  The League was flawed from the start - without every powerful nation being a member it would lack importance.
During key crisis, especially Manchuria, this became critical. In Manchuria America, despite isolationist polices, was involved in China and thus had some say in the Far East. Had America taken a very strong stance towards Japan through the League in the early 1930s, Japan would be put in its place and it would be a huge, confidence boosting victory for the League of Nations.
However, President Hoover probably wouldn't have done a lot, i.e. not employ sanctions on Japan, simply because of the Great Depression.
Even the economic sanctions imposed during the Abyssinian crisis were unlikely to succeed without America. America was under no obligation to employ sanctions as it was not a member of the League, so it continued to trade with Italy.
Russia (USSR) did not join until 1934. Though the USSR was seen as a pariah, if it had joined the League at the start, the League would have had more status. Not a huge factor, but still worth noting.
Had USA, Germany and USSR joined in the start, the League would perhaps have been more successful. (But probably not have saved it).

Britain and France were often unwilling to get involved in League affairs, preferring to make sure their national interest was defended. Britain and France were the head and shoulders, with Japan and Italy below them...
Victors club. Lack of participation also meant that the League was dominated by Britain and France. They tried to turn it into the type of organisation that suited them. America perhaps would have curbed this, perhaps trying to include the smaller countries more. When Germany was allowed to join in 1926, the idea that the League was a 'victor's club' was made clearer as Germany never had any real influence… Hitler walked out in 1934. (Same applies with USSR not joining until 1934)
Imperial and domestic problems of B and F dominated the League. It often seemed that the aims of the League were what the victors want. For example France with the Rhur.
The French basically only cared about punishing Germany. Britain was more focused on peace but also conciliation Germany. Conflict of interests!
Britain and France were in fact on the decline! Russia, Japan and  Germany were, rather, on the way up.

The covenant was too vague. There was no specific structure, especially for dealing with non-members. It relied on the 'good will' of members too much.
No army! With member states unwilling to provide soldiers, the LON never had an army - its only weapon was sanctions. These sanctions were limited with America not being a member of the League, and any sanctions employed were partial only (Abyssinia).

Japan, then Italy were able to get away with invasions of other countries.  This made the League appear powerless.  This meant Germany could consider breaking the Treaty of Versailles.
Manchuria. This was the first 'big test' for the League… it failed miserably! Lytton Commission was an inadequate response and showed just how weak the League was. A lot of talk not a lot achieved…
Japan left the League. This further showed how weak the structures of the League were.
Abyssinia. The final nail in the coffin. Half-hearted, weak sanctions showed how little muscle the League had. The return of realpolitik and the Hoare-Laval plan showed that Britain and France were not fully committed to the League (also shown with Munich conference). 

Expectations were too high. The reality was that the League was never going to succeed. In the post-war 1920s it could partially function due to good will.
The League relied on disarmament. Member states simply didn't trust each other! Who would disarm first? BUT after the impact of the Great Depression and the rise of extreme right-wing government in Europe and Asia, it had no chance of success.