arm in a sling

今日のジョークは私は余り笑えなかった（つまり意味が分からなかった？）のですが、まず読んで下さい。
A defendant appeared before me in the magistrates' court, charged with stealing two washing machines. He was covered in bruises and had one arm in a sling.
"My client wished to apologise for his appearance," explained the defence solicitor, "but he fell off the back of a lorry."
謝ったことが洗濯機を盗んだことではなく、怪我をしてみっともない姿だったのがジョークになったと解釈していますが、この解釈には自信がありません。何か他の解釈はあるのでしょうか？私としては面白いジョークとは思えなかったのですがこのジョークを取り上げたのは "arm in a sling" の表現を覚えたかったからです。
"sling" を辞書で引きます。
OneLook Quick Definitions:　bandage to support an injured forearm; consisting of a wide triangular piece of cloth hanging from around the neck
Dictionary.com:
・a bandage used to suspend or support an injured part of the body, commonly a bandage suspended from the neck to support an injured arm or hand.
・a strap, band, or the like, forming a loop by which something is suspended or carried, as a strap attached to a rifle and passed over the shoulder.
"arm in a sling" は腕を骨折した人が三角巾で腕を吊る姿ですね。