The dative sentence is wrong. That письму is 'to (a) letter", and меня is me in accusative, so that sentence is "Give me to the letter". "Give the letter to me, me is dative, letter is accusative. Письму is the dative form though, I believe.
I'm pretty sure через matches with accusative, not instrumental (Which in most cases, means it will look like it matches with nominative). "I replied by letter" would (non native here) be more like Я ответил по письму.(that being dative)
For instrumental, you'd need words like под, or над. Exposure to russian writing will sort everything out over time. Eventually you see the word над enough, until you know which case it matches with (which varies depending on the situation). In a large number of situations, if a preposition is used with a verb (Which is arguably rare with russian), нa + prep is the way to go. Because the key of the whole system, is that there are no "floatin word". No word is just... There. Every word has it's reason appointed, and is connected logically to something else. Keep in mind, various propositions have different meanings when used with different case. Mainly, с(о). It could mean "with" (+inst), or mean "from" (+ gen), when talking about motion from somewhere, as opposed to "from" as in "origin". Motion generally changes the case. Под + inst= under .... Под + acc= to under, moving under.

Also, just keep in mind, everyone will tell you there are 3 genders, about three declensions, and six cases. Which is certainly enough to get by understanding. But also note that words like время and имя are not the same as лексика in any way, just because they all end in an a sound. Generally, prepositional covers the whole deal, but in some words, their system is split into Nom, Acc, Gen, Dat, Inst, Prepositional, and locative. As well as certain words also can be found in a "partitive" case (which remarkably resembles genitive in many ways, but is a little different). Exposure teaches everything over time.