うめえ。 5キロといわず もっといけたな
The sentence before this has nothing to do with it (the character is eating and not listening to the other character's speech) so he blurts out this.
The only context is that he asked 5 kilos of meat to eat so they made a dinner for him with 5 kilos of meat.
といわず seems to be "Whatever is or not..." but then how does it related to もっといけた? Which seems to be "It's more delicious"?
The unrepeated ～といわず is listed as #1 definition in this デジタル大辞泉 entry:
と‐いわず〔‐いはず〕【と言わず】 の解説
［連語］
１ …と言うことなく。…などと言わないで。「明日といわず今すぐ」
２ （多く「…といわず…といわず」の形で）特に…と言うことなく全部。「顔といわず手といわず泥だらけだ」
And #2 definition in this プログレッシブ和英中辞典 entry:
－といわず【－と言わず】
1〔…やら〕
頭といわず，背中といわず，ばんばん叩いた
I hit him on the head, on the back ― everywhere.
2〔…と言うことなく〕
明日と言わずに今日やってしまいなさい
None of this “tomorrow” stuff ― get it done today!
The といわず in your example means ～などと言わないで, but I think it has a nuance of "not just... / not only... / without limiting it to...", and when used in this sense, I think it's often used with a phrase containing a numeral, eg 「３日といわず、もっといてください。」(Please stay longer, not just 3 days.)「１個といわず、５個でも６個でも好きなだけどうぞ。」(Don't just take one. Feel free to take five, six, or as many as you like.")
いける here means "can eat" ≈ 食べられる. eg 「わんこそば100杯いける。」(I can eat 100 bowls of wanko-soba.) 「もひとつ、いける？」「もう無理。」(Can you eat one more? -- No, I can't.) I think this usage is quite colloquial and a little slangy, so it's not listed in 明鏡国語辞典 or デジタル大辞泉. The closest that I've found online is 「ご飯3杯はいける I can eat 3 bowls of rice with this.」 on Weblio 英語表現辞典.
So your example can be rephrased like this:
うめえ。5キロといわず、もっといけたな。
≈ うめぇ。5キロなどと言わないで、もっと食べられたな。
"Delicious. Not just 5 kilos, I could have eaten more."
It would appear that a phrase has been suppressed here, since といわず occurs in pairs, usually:
日本語といわず外国語といわずどんな言葉でもそうなんです "It's the same in any language, whether it be Japanese or a foreign language."
先生といわず医者といわず考えはおなじでしょう 'Not only the teacher, but the doctor too thinks the same way, you see.'
So I would imagine the sense here is '[I] could have done with more, not only 5 kg'. This is 話し言葉, so my understanding may be faulty; others will correct me, if so.
