Let us start with the meaning of the verb 通す here, and work our way up to the whole sentence. We will touch on a few incidental points of interest along the way.
The meaning of 「通す」
"Stick with [something] (until the end/all the way)" is indeed one of the meanings of 通す, but in this context it means "go through [something]", or to be more specific, something along the lines of "use the mediacy or mediation of something/someone" or "use something/someone as a medium or intermediary". The most relevant definition on goo辞書:
Thus 「麻雀を通す」 may translate to "use mahjong as a medium".
Why is the verb in て-form?
In your question, 通す is in the non-finite te-form 通して. The te-form has a wide array of uses, one of which is to indicate that the action expressed by the verb is a means for doing some other action. This can be represented in English by "by Verb-ing" or by turning the verb into the participial form (i.e. "v-ing"), which is also a type of non-finite form.
So "(by) using mahjong as a medium" is an apt translation for 「麻雀を通して」.
A syntactic analysis of 「を通して」
You can think of "を通して" in your question as, rather than expressing a distinct action, acting more like a case particle that serves to indicate that (the referent of) the preceding noun is a means or medium through which the event or action expressed by the following verb phrase or clause is achieved. In cases like this, it might be better to translate it as "through", "by the medium of", "by means of" and the like.
This is in the same vein as the likes of "に対して" ("toward" or "against"), "について" ("about"), "をもって" ("with") and "によって" ("by" or "depending on"), all of which are originally comprised of a particle and a verb in te-form but have gone through a conversion into compound particles.
「なら」: The grammar
If it were 「麻雀を通すなら」 or 「麻雀を通したなら」, the 「なら」 would be a 接続助詞 ("conjunctive particle") like 「ので」, 「から」 and 「 けど」 etc., but the 「なら」 in 「通してなら」 can be considered a 副助詞 (What's the English term for this one?), which puts it in the same class of particle as しか, こそ, でも, etc., and as such, it can attach to various kinds of part of speech. For example:
「彼女ならできる」(attaching to a noun)
「田中さんになら話してもよい」(attaching to a particle)
「ゆっくり(と)なら進むことができる」(attaching to an adverb)
「なら」: The meaning
To investigate what the なら does in terms of meaning, let us first look at the なら-less version of the sentence, 「麻雀を通してお姉ちゃんと話せる気がする」, because it is already a grammatical sentence without it.
As you probably understand now, it may translate to something like this:
"I have a feeling that I am/will be able to speak with my sister through mahjong."
(I used "through mahjong" for 「麻雀を通して」 here but I think "(by) using mahjong as a medium" is fine too.)
By attaching 「なら」 to 「麻雀を通して」, you are adding some import with respect to mahjong as a means of communication.
So what is it?
The more obvious answer is that the 「なら」 indicates that "through mahjong" is the condition for the speaker's being able to speak with her sister. This is a possible interpretation. Indeed, in 「ゆっくりとなら進むことができる」, for example, "slowly" is a condition for being able to proceed.
The less obvious, but more accurate answer here, I think, is that it ascribes a degree of specialness or exceptionality. Let me use the example sentences in the previous section to illustrate this.
This means "She can do it," but with an implication that she is somewhat special or exceptional in being able to do it.
This means "You may talk to Tanaka-san," but with an implication that 田中さん is something of an exception.
(Compare this use of 「なら」 with the 副助詞「だけ」, which (among other functions) indicates a sole exception.)
A general look at the sentence
Given all this, we should now understand that 「麻雀を通してなら」 in 「麻雀を通してならお姉ちゃんと話せる気がする」 carries the information that using mahjong is a special or exceptional (in the sense that there are only a few other ways, if at all) means of achieving a conversation with her sister.
Now, I would provide a translation for the whole sentence by way of a closure, but I am having a hard time fashioning one that fully and neatly incorporates the implication of 「なら」. Perhaps fronting through mahjong helps a little?
"I have a feeling that, through mahjong, I am/will be able to speak with my sister."
Well, anyway, this concludes my overlong answer.