Although it is a lengthy phrase,
Intrigued at the idea of putting a general in charge...
is not a dangling modifier, as it does immediately precede Mr. Trump, the noun it is modifying, as @FumbleFingers indicated in his comment. A dangling modifier is present in this variant of the sentence:
Intrigued at the idea of putting a general in charge, the job was offered by Mr. Trump to Mr. Kelly a few days ago.
or even worse,
Intrigued at the idea of putting a general in charge, Mr. Kelly was offered the job by Mr. Trump a few days ago.
So what is it? I would classify it as a participial phrase, where the entire phrase is an adjective modifying Mr. Trump. Specifically, it modifies his state of mind, or intent, for offering the job to Mr. Kelly, as pointed out by @Tᴚoɯɐuo.
Finally, "intrigued at" appears to be the least common use among "intrigued by," "intrigued with," and "intrigued at." However, all three could be considered valid uses.