Multiple introductory phrases and commas

The rule says that a comma will be required whenever the normal word order is disrupted. Except for adverbials of time, which can and usually do appear at the beginning of the sentence, all other adverbials will usually come after the objects (if any) within the predicate.

The presence of "with a fast space" (adverbial of manner) in the first sentence forces you to use a comma. Now, let's agree that the normal word order would be:

• In the morning(,) I will get up at a fast pace. (I'd use "at", not "with".)

When we have two or more adverbials of the same kind, we usually order them from more particular to more general (in which case no commas will be required). If that order is disrupted, then a comma will be needed. In your second sentence, we must assume that the little compartment is found in the table, rather than the other way round, so the normal position of those adverbials, from more particular to more general, should be as follows:

• In the little compartment (which is) under the table there is a key.

A comma could be used to render a non-restrictive meaning, as if it were an afterthought or a clarification:

• In the little compartment, (the one that is) under the table, there is a key.