It seems that 《說文解字》 does not have the entry of 奈.
奈 was derived from 柰. It might be a new word, or due to miswriting. The original meaning of 柰 is the name of a fruit. 柰 is then borrowed and used in 柰何, which means 如何 (how; what; what way). Later, in most cases, 柰 represents the fruit name, and 奈 is dedicated in 奈何 and the like words.
① <副> 无奈。 (however; how can one help; how to handle it)
② <动> 通“耐”。禁得起；受得住。 (to stand; to bear; to endure)
③ <动> 处置；对付。 (to deal with)
According to this, the 奈 of 无奈 is an adverb, and means "how to handle it" or "with means". Then, 无奈 means "not to know how to handle it" or "without methods".
① 谓无可奈何。 (cannot help it; to have no alternative; to have no choic)
② 犹无比。 (incomparable; matchless)
③ 犹可惜。 (unfortunately; It's a pity.)
无奈 may mean 无可奈何 or others. That is why we have two phrases instead of only 无奈. We also prefer different words/phrases to prevent repeats.
① 怎么，为何。 (why)
② 怎么样；怎么办。 (how; how to do)
③ 犹言办法。 (method; means; way)
④ 谓采取手段、办法整治对方。 (to adopt means or methods to handle others; to do something to others)
Depending on the context, 奈何 may be an adverb (① and ②), a noun (③) or a verb (④).
If 无可奈何 is a brief term of 不知道该怎么办或不知道如何处理 (do not know how to handle it)，奈何 is ② an adverb phrase.
If 无可奈何 is a brief term of 没有可以用的办法 (there is no way to use it), 奈何 is ③ a noun.
If 无可奈何 is a brief term of 没有办法可以奈何得了他／它 (there is no way to treat him/it...), 奈何 is ④ a verb phrase.
无奈我爱你 cannot be accepted as "I can't help to love you".
"I can't help to love you" means 我不能帮着爱你.
"I can't help loving you" means 我不能不爱着你, 我没有办法停止爱你 or 我禁不住地爱着你.
无奈我爱你 may mean 没有办法（改变），我就是爱你 ((there is) no way (to change), I just love you).
Although it is not so common, we can still use 有奈. 有奈 is intentionally used as the antonym of 無奈. Thus, they are often used together. Otherwise, many people do not know what 有奈 means.
「有奈無奈，赤腳跑到五寨」是清朝 康熙年間，雲南 鶴麗鎮 總兵 郝偉（1653—1718）的發跡故事 (the story of 郝偉)。
「变无奈为有奈，化奈何为何奈」是一篇文章的標題 (the title of an article)。
「因為無奈 台灣現在能有奈嗎？」是一則新聞的標題 (a news title)。