be on (one's) last legsは、疲れて(年老いて)今やっていることを続けられない、legsには持続力とか、グラスの内側に垂れるワインの筋という意味があります。
その老人は死にかけていた。→The old man was on his last legs.
Nakagawa debackle pushes PM to brink; ongoing wrangling stealing focus away from economy.
The resignation of Finance Minister Shoichi Nakagawa - who gave an embarrassing performance in Rome after a Group of Seven meeting earlier this month - has apparently pushed the cabinet of Prime Minister Taro Aso to the edge of cliff.