How connected are information theory and algorithmic information theory?
I don't have the background to give a deep answer about how deeply connected the fields are, but Li and Vitanyi's An Introduction to Kolmogorov Complexity and Its Applications--probably the most widely known textbook on AIT--repeatedly incorporates mathematical and conceptual material from information theory (such as Shannon entropy) into its presentation of AIT.  Here is a quotation from the third edition (p. 73) that illustrates a connection that might be considered deep:


  We are interested in a measure of information content of an individual
  finite object, and in the information conveyed about an individual
  finite object by another individual finite object. Here, we want the
  information content of an object x to be an attribute of x alone, and
  not to depend on, for instance, the means chosen to describe this
  information content. Making the natural restriction that the
  description method should be effective, the information content of an
  object should be recursively invariant (Section 1.7) among the
  different description systems. Pursuing this thought leads
  straightforwardly to Kolmogorov complexity.


(Apart from the fact that most of the paragraph clearly seems to be discussing Shannon information, it comes immediately after a paragraph that discusses a quotation by Shannon.)