By no means has he promised to keep you informed of how their business was going on. <<<
By no means, he has promised to keep you informed of how their business was going on.
Under no situation should I do that. = I shouldn't do that under any situation.
In no case would he be that bad. = He wouldn't be that bad in any case.
By no means是一个介词短语，在整句中起到副词（组）的修饰作用，原句主语是 he。谓语部分是 has promised. 及动词为promise。“Promise to do something”，一个很普通的结构，相信你不会感到陌生。
Keep you informed = I'll always inform you, I'll let you be informed 就比如 Keep that thing good, keep her happy, 及 keep + 宾语 + 形容词 的结构，将某物保持在某种状态。
I'm informed. 我被通知了，我被告知了。
I'm informed of that story. 我被告知了那个故事。所以介词of在这里起到“桥接”被动状态下的动词与宾语的作用，算得上是一种“双宾语”， You inform me the story 或者 you inform me of the story. 也可以说 I'm informed that + 从句。
of how... 被人叫做“介词从句”。