First of all, I would like to suggest that the question should be: Does this sentence make sense? 我打算工作在一个网站会完了在四月十五, because your concern is "Even though I feel like my version is wrong, can someone point out why?"
我正在建立一个网站,应该在 4 月 15 日之前完成。This is almost perfect, we would not discuss it further.
我打算工作在一个网站会完了在四月十五 this sentence has a couple of questions that need to discuss.
(1) 我打算 and 我正在 are different meanings. You decide which one you want. If you already started the project but have not finished yet, I suggest you say 我正在.
(2) 工作在一个网站 is not a correct usage of 工作. It is usually used as 在一个网站工作, which is not what you mean. You'd better use another verb 做 or even better 建立. In Chinese, you need to be careful when translate "work on something" (做) and "work for somebody" (工作).
(3) 会完了在四月十五 would be better structured as 会在四月十五完成。In Chinese, an
adverbial phrase of time goes before the verb 完成. Here you definitely do not want to use 了 because this word means you already get the job done.