"At all" vs "Not at all" in negated sentences

If I say That makes sense to me. I would say definitively at all.

That makes sense to me at all.

But in the negated sentence I'm not sure. I've the feeling, that it is still at all. But if I say not at all it still sounds OK to my ears but I would, though, separate it with a comma or even with a full stop.

Which one is correct?

That makes no sense to me at all.

or

That makes no sense to me, not at all.

That makes no sense to me, by no means at all.

At all is a Negative Polarity Item (NPI). That means it is one of a long list of English words, phrases, idioms, and constructions that can only occur grammatically in a Negative environment.

Rather than being "awkward" or "ugly" (which are esthetic judgements and not grammar),

• *That makes sense to me at all.

is simply ungrammatical, since there is no negation present to license the use of at all. So it's incongruous here; what could it possibly add to the meaning in such an environment?

In a grammatical sentence, say,

• That doesn't make sense to me at all.
• That makes no sense to me at all.
• Nobody understands that at all.

at all functions as an intensifier of the negation, meaning something like 'in even the most insignificant manner'. But without a negative environment to license it, at all makes no sense at all.

There's a lot more to say about Negation and Negative Polarity, but this is a very simple case.

Edit: Specifically, just to get it all on the record, for the future, here's the list of Negative Triggers from http://www.umich.edu/~jlawler/NPIs.pdf. (If-Clauses fall under "III. Hypothetical Clauses)

Negative ‘Triggers’

Notes:

• NPIs are illustrated here by the prototype English Negative Polarity Item any.
• *A asterisk before an sentence means (that) the sentence is ungrammatical.
• Parentheses indicate optional material (that) one can either omit or include.

• The first few sentences give grammatical negatives versus ungrammatical affirmatives. After that you're expected to find your own ungrammatical affirmatives. (It's rather fun, actually; they really sound terrible)

• I. Overt negatives:

  • A. not [w/ NPI clausemate or complement]

  • *He (did know/knew) anybody.

  • He didn't know anybody.
  • *He (did claim/claimed) (that) he knew anybody.
  • He didn’t claim (that) he knew anybody.
  • *He thought (that) he knew anybody.

  • He didn’t think (that) he knew anybody.

  • B. Incorporated negatives

  • I doubt (that) he knows anybody.

  • It’s unlikely (that) he knows anybody.
  • It’s improbable (that) he knows anybody.
  • He dislikes anybody(’s) reminding him.
  • He prevented her (from) seeing anyone.

  • I kept her from telling anybody.

  • C. Negative frequency adverbs

  • I seldom/rarely see any of them.

  • He hardly/scarcely knows anyone.

  • D. Quantifiers & quantified adverbs

  • Only Bill did any of the homework.

  • Few people see any use for it.
  • *A few people see any use for it. (few is negative; a few isn't)

• II. Questions (overt and embedded)

  • Did you see anybody?
  • How does anybody stand this?
  • I wonder how anybody stands this.

• III. Hypothetical clauses.

  • Tell me if you see anyone.
  • He asked whether I saw anyone.
  • Tell me whatever anybody says.
  • Check the list again, lest we forget anyone.
  • I’ll read it aloud, unless anyone objects.

• IV. Comparatives, superlatives, etc.

  • There’s more/less here than anybody knew.
  • He prefers beer to any other drink.
  • He’d rather die than hurt anyone.
  • He’s as good as anyone expected.
  • He’s the fastest one (that) anybody knows of.
  • I saw him before anybody (did).
  • I’m surprised (that) he knows anybody.
  • It’s too dark to recognize anything.
  • He left without anybody noticing (it).
  • That’s hard/tough for anybody to do.

OP has usage of "at all" the wrong way round - we only normally use this in the negative.

The only really sound example sentence is That makes no sense to me at all, but I think even that would probably normally be phrased That makes no sense at all to me.

*That makes sense to me at all is definitely unsound (but That all makes sense to me is okay).

?That makes no sense to me, not at all. Some may find this acceptable, but I could only endorse either That makes no sense to me, none at all, or perhaps That does not make sense to me, not at all.

*That makes no sense to me, by no means at all is also unsound. Primarily because of the tautology of by no means and [not] at all, but the double negation no sense/no means is also ugly.