Let it be done - grammatical analysis

How should the phrase "Let it be done" be interpreted grammatically?

What confounds me are the following assumptions, some or all of which may be wrong:

1. "Let" implies imperative
2. "it be done" implies a subjunctive
3. the lack of an agent implies passive
4. section 5 of this reference says that the "let" indicates the desire for a hypothetical situation and is hence part of the subjunctive - I'm thinking that this "let" could be either a desire and a command but at this point I'm probably wrong

Given the above is "Let it be done" a correct passive transformation of "Do it"? Is this a passive-voice imperative-mood construction (in which case what about the subjunctive?), is it a passive-voice subjunctive-mood construction? Is it somehow in the active voice ?

The original query was intended for "Let the cake be eaten", which I had assumed to be a passive transformation of "eat the cake" the title and rest of the question are an attempt at generalization. Also note that I'm looking at this from a purely grammatical point of view, whether such constructions are silly or not is another matter.

related questions here and here

Nordquist has a balanced article on jussives that examines different approaches.

I'd say that calling 'Let us pray' an imperative usage is stretching the term somewhat. It is far less hortative than '[Get] on your knees!' In the linked article is:

'[John] Lyons [Semantics, 1977: 747] argues that the imperative can only be, strictly, second person, and never third person (or first person). _This may, however, be no more than a terminological issue, since first and third person 'imperatives' are often simply called 'jussives.'

Types of jussives are also addressed:

Jussives include not only imperatives, as narrowly defined, but also related non-imperative clauses, including some in subjunctive mood:

Be sensible.

You be quiet.

Everybody listen.

Let's forget it.

Heaven help us.

It is important that he keep this a secret.

...................

In 'Let it be done,' 'it' is referential (with preceding referent) and 'be done' is certainly passive. A modern not-too-near paraphrase is perhaps 'Get on with it, or make sure someone else does!' Far more imperative.