I wanted a sentence using 寝/ね.

「寝が足りない」って、正しい表現のようですが、恥ずかしながら初めて目にしました。

ぼくは「寝足りない」と言います。こちらの方がよりよく使われるようです。
[#3060523] まだ寝足りないでしょ？
[#3227545] 最近寝ても寝ても寝足りない。

https://www.google.co.jp/search...86.gqcq7XzVdSw
https://www.google.co.jp/search....0.urjWby42Xhc

I was hoping to be able to have a sentence where 寝 was being used as a noun by itself. 寝足りない seems to me to part of the compound verb 寝足りる, which can be found in the 新和英大辞典第５版 (ねたりる【寝足りる】...). That dictionary uses 寝が足りない as an example of 寝, so I thought it would be safe as a sentence. It also uses "寝が足りないのでだるい". Would that be better?
I'll certainly use your sentences containing 寝足りない as examples for 寝足りる.

I didn't mean to say this sentence was wrong. I can see on the Internet that 寝が足りない is actually used, so I agree that there should be a sentence here using it. I just though it might be a good occasion to show the limits of my "I wouldn't say it", and I thought it might be nice if someone who happens to visit this page can also learn the expression 寝足りない, which seems to be more common than 寝が足りない.

I'm not sure if this sentence sounds very natural, though. It's not easy to make a good example sentence from a verb phrase. It might sound as unnatural as あ、僕はカメラを忘れた！ (cf. [#2515530]).
I'm not sure if 寝が足りないのでだるい。 is better, either. I can't find any actual usage on Google using this pattern. Although I'm a great fan of 研究社和英大辞典 myself and I always use it when I write English, its Japanese sentences don't look perfect to me. There is no grammatical mistakes, of course, and I believe the translations match very well, but they haven't been proofread from the perspective of Japanese-language education, have they?

Something like ああ、寝が足りない。 might be better than this (cf. http://d.hatena.ne.jp/kaerudayo.../20071011#p2).
Or maybe you could use one of these sentences if you think it's safe enough.

http://blog.goo.ne.jp/renka_k/m/200904/1
寝が足りないので、今日は早めに寝ます。

http://blog.goo.ne.jp/ohkubo-ta...be26dbfb2859d3
昨夜は４時まで起きていたので、少々寝が足りない。

Yes, 寝足りない is FAR more common - the Google n-gram corpus has it about 40 times more often than 寝が足りない.

I like the "寝が足りないので、今日は早めに寝ます。", and I think I'll add it. A point on translation; literally it says "today I'll go to bed early", but in English we'd usually say "tonight I'll go to be early". I don't want to get 直訳, so I'm inclined to apply 意訳 and use "tonight". Would 今晩 be possible there instead of 今日?

Interesting your comment on the big 研究社. They have heaps of Japanese editors, and they claim to try and make everything sound natural.

はは、無名の若造が天下の研究社にもの申したところでどうしようもないのは分かっております。

今晩 is possible. 今夜 might sound better, but 今日 seems to be much more common.
If "tonight" sounds more natural in English, I think it's no problem to have a pair using 今日 and "tonight".

I've made it "tonight".