The 33-year-old midfielder - the Hammers' captain until he left the club by mutual consent just days before the conclusion of the summer transfer window - will remain with the League Two team for an indefinite period.
"It is great to have such a model professional in Kevin at the club, and I am more than happy for Kevin to come in and train with the squad," Orient manager Ian Hendon, who has inspired his side to first place and five wins from their opening six league fixtures this season, told the club's official website.
"It is good for our players to be training with a vastly experienced and talented Premier League midfielder."