What's to become of Mike now, with his father and mother gone so quick, and nobody to keep him in check.

1. What's to become of Mike now：按系表结构理解。become of sb. = happen to sb.

2. with his father and mother gone so quick,  是with 复合结构。构成：with +宾语+adj。quick 用作副词，同quickly。

3. and nobody to keep him in check，在and 后省略 with。也是with 复合结构。构成：with +宾语+ to do。

【翻译】现在迈克会怎样呢？他的父母那么早就双双离世，又没有人约束他。