In? In what? In traveling or "in administrative details"?

No, it means that the time was spent either in traveling within the United States, or in travelling to and from the United States. "In" travelling.

I agree with you but it is poorly written not that I am anyone to be able to complained about poorly written sentences.......LOL

In means in the States (USA). So he spent most of his time in America or travelling to and from America. He may or may not have been travelling around America, he may have just stayed put.

Any errors are due to me, not of the author, the prestigious John Leighton Stuart.

A typo there: It should be "Peking," not "Peing."

So I think the States may be referring to the provinces of then China.

I am pretty sure he meant the United States. In 1919 John Leighton Stuart became a professor at Yenching University. Shortly after the establishment of the People's Republic of China in 1949, Yenching University ceased to exist. In 1952, Peking University relocated to the Yenching campus and absorbed most of its academic departments while other departments were absorbed by Tsinghua University.

He had links with academic institutions in the USA, and later became US Ambassador to China.

As for your question, my take on it is this:

But I ... soon began to spend much of the time in [or traveling to and from] the States

In the USA or travelling to and from the USA.