If you are only 2 subjects (Alex and her), where Alex is waiting for her for 30 minutes, or Alex has been asked to wait for her, the sentence is simple :

Elle fait attendre Alex depuis 30 minutes

You can’t use “à nous” at the end of the sentence, I only see one way to say it :

Elle nous fait attendre depuis 30 minutes

If Alex is waiting for us because of her, it becomes a quite complex structure, but you should say :

Elle fait Alex nous attendre depuis 30 minutes
Elle le fait nous attendre depuis 30 minutes (nicer)

The problem is that the sentence is odd and such a structure is very complicated to understand while speaking (in written expression, you can read 3 time the sentence, there would not be any problem, but in oral expression, you won’t make the person repeat 3 times…).
So you should say :

Alex nous attend depuis 30 minutes à cause d’elle

For you last exemple :

Tu as besoins de leur faire faire le ménage

is not weird. We often repead twice “faire”, so we understand it easily.

Note that the structure «Il me fait faire le lavage» is not weird but we don’t say “lavage”, we would say “la lessive” (clothes), “la vaisselle” (dishes), “la toilette” (shower)…

To use this construction without a pronoun you should use the following for intransitive verbs (those used without objects):

Elle fait attendre Alex depuis 30 minutes.

Elle fait sortir Pierre de la pièce.

With transitive verbs you can add the real subject as a complement introduced by à or¹ par.

Elle fait payer le resto à Alex.

Elle fait ouvrir les huitres par son cuisinier.

If the subject or the object of the construction are pronouns, there is no other choice than placing them before the conjugated verb.

• Subjects of intransitive verbs become direct object pronouns:

  Elle nous fait attendre.

  Elle le fait sortir.

  Elle les fait se reposer.

• Subjects of transitive verbs become indirect object pronouns:

  Elle me fait payer le resto. / Elle me le fait payer.

  Elle lui fait payer le resto. / Elle le lui fait payer.

  Elle leur fait faire le ménage. / Elle le leur fait faire.

Note: For emphasis is however possible to use a preposition with emphatic pronouns in some cases, but the pronouns need to be repeated:

Elle nous fait attendre, nous !

Elle lui fait ouvrir les huitres, à lui !

And indeed “faire faire quelque chose à/par quelqu’un” is perfectly correct for “make/have someone do something” and it is often used in French.

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^{1. Subtleties apply, cf. last remark here, it roughly corresponds in English to the nuance between “make someone do something” or “get someone to do something” and “have someone do something”.}