With respect to the first question:
First, the choice of tam...ut for the structure looks very apt. Now, for the second part ut dormiendus eam, few things ought to be said: (1) It does not mean the required since it sets apart the passive adjectival "dormiendus" from the verb eo, so as it stands it could not mean "should go to sleep"; It might be the case that the intention here was to use the accusative supine dormitum which is indeed an idiomatic option here (both eo cubitum and eo dormitum are attested). But (2) having the plane ut dormitum eam does not provide us with the flavor of "should" or the judgment here, so I guess that was the idea of using the gerundive. I've once had this contemplation on how to render "should" and I was also considering a gerund/gerundive. However, the gerundive is usually much more forcing and stronger than the "should" here. The answer there suggests using debeo which is a good option. Alternatively, in this case, we could also use the impersonal oportet. So we can say: Tam/adeo fessus sum ut ire dormitum debeam.
[Had we wanted to use the gerund/gerundive anyway, we could say something like this ut dormiendum mihi sit - we could not have used the gerundive dormiendus because it is passive; i.e., if we say dormiendus sum it means "I must/should be slept"]
About the Second question, I believe it merits its own question; So briefly, you are correct in your judgment: libri legendi sunt means the books must be read; If you want to signify the purpose of the books there are several options and most naïve one is probably ad legendum. Amica mea libros emit ad legendum.