How did the League respond to the abyssinian crisis?

Fairly poorly is the simple answer!

The LON did not want to anger Italy, however their invasion went against their policy of collective security, and so they condemned Italy's actions, and imposed a trade ban. However this trade ban did not include fuels, such as petrol and oil. If there was no ban on petrol and oil, Italy could still continue their localised war against Abyssinia.

To try and appease all those involved, the Hoare-Laval Pact was set up in 1935. Within this pact, the French and the British designed a compromise; Abyssinia would be split in two, the wealthiest half given to Italy. However there was a leak of this information to the press, which resulted in uproar amongst the British public, as it appeared Britain and France were surrendering to Italy's aggressive actions.

Italy, following Japan's previous example in the Manchurian Crisis of 1931-1933, left the League. After this major failure, the League was no longer taken seriously.