Something to think about:
Consider the number
It is highly determined, in fact one can come up with a formula for the n-th digit.
Given any number, it is obvious that it appears infinitely many times in this decimal.
It is conjectured that pi has the same property; it is known that almost all real numbers do.
Dr S: Anything to add to AG's assertion? or mine?
It's possible to generate a system that nearly "appears random." That's what Pseudo-Random-Number-Generators do. "Random" is more appropriately applied to a process, not the result of such process.
No written string of numbers is "random"; it merely is. The best one can do is say that a string of numbers obeys certain laws that randomly generated strings do.
Champernowne's number (.12345678910..., as you gave) does have the property that "any string occurs with the proper frequency." Thus (if mapped to a alphabet), it would contain the complete works of Shakespeare, the complete works of Shakespeare with one error, etc. (and the complete text of "Contact.") However, it can be proved (somewhere, I don't have access to review journals) that Champernowne's number does not obey the Law of the Iterated Logarithm. (I don't know how to generate a number that does except by ad hoc post hoc adjustments to the output.)
I see that Doctor Stochastic has already replied, so all I shall do is add a link for Lurkers to meditate on the difference in auto-correlation between a decimal based Champernowne's constant and a binary: from Mathworld
Beyond that, I suspect the more direct approach to acheiving a compilation of Shakespeare's works is to digitize the same and merely count up to it. LOL!