Sleeping 是指猫的动作Jane will have 是定语从句修饰pets 定语拿掉就变成the cat is the only one of her pets sleeping on her bed 。sleeping on her bed 分词短语修饰cat。

表示状态呗
The cat is the only one of her pets ((that) Jane will have) sleeping on her bed.
括号里是个从句 先行词省略了
这句子怪怪的 应该是为了出题硬挖的坑 一般没人会这么说话..