[Sashimister]

５１）彼等はその年老いた母と一緒に、三人で此所に来 て居た。
They had come to this place as a trio along with an elderly lady.

"Mother" instead of "lady"?

５２）いろいろな反覆する機会からして、避けがたく私 はその女づれと懇意になつた。
Through various reoccurring chances, the hard-to-avoid me and those girls got to know one another. (yeah, definitely got confused on this one. I assumed 避けがたく was modifying 私）

避けがたく modifies 懇意になつた. ", I unavoidably got acquainted with the girls."

Adjctives ending in く can only modify verbs.

５３）遂には姉娘と私だけで、森の中を散歩するやうな 仲にもなつた。
Finally, I and the elder sister alone, came to have a relation like walking through a forest.　

"relation like walking through a forest"?

Maybe something like "got close enough to take walks through the forest by ourselves"?

５４）その年上の女は、明らかに私に恋をして居た。彼 女はいつも、私のことを『若様』と呼んだ。
It was obvious that this somewhat older girl was in love with me. She always called me “young master”.
Good.

５５）私は最初、女の無邪気な意地悪から、悪戯に言ふ のだと思つたので、故意（わざ）と勿体ぶつた様子など して、さも貴族らしく返事をした。From the outset I had, from her innocent mischief and because I thought she was talking teasingly, purposefully done things such as put on airs and in that way answered her in the manner of an aristocrat.

Not sure if you got the structure. The first half of the sentence refers to the fact that she was calling him young master.

"First, I thought she was just playfully calling me that from her innocent mischief, so I deliberately reacted by putting on airs and replying like an aristotrat."

５６）だが或る時、彼女は真面目になつて話をした。
However, now and then she would become earnest and talk.

"at one time, she talked to me in a serious manner."

５７）ずつと前から、自分は一高の運動会やその他の機 会で、秋元子爵の令息をよく知つてること。

Since a long time before, I had, though Tokyo University athletic meets and other circumstances, known of Count Akimoto’s son.　(I, or she? I wasn't certain about that 自分).

自分 = the girl. 自分 modifies 知つてる. I don't want to start lecturing but 自分 is one of the key words to make your Japanese sound more natural. Seems to me Japanese-learners seldom use it. We use it so many times daily.

I don't think the hero even knows of Count Akimoto's son to begin with.

５８）そして私こそ、たしかにその当人にちがひなく、 どんなにしらばくれて隠してゐても、自分には解つてる といふことを、女の強い確信で主張した。　And I, who was certainly not this person, no matter how much I feigned ignorance to hide it, she came to a conclusion by herself and insisted upon it with firm conviction.

You probably paid too much respect to the original Japanese word order, resulting in a hard-to-follow translated sentence. To me, yours starts off sounding like that "I" is the subject and then it changes to "she" in the middle.

I would do it as "With a woman's firm conviction, she insisted upon knowing that only I could be the person in question even if I would feign ignorance and hide the fact."

５９）その強い確信は、私のどんな弁駁でも、撤回させ ることができなかつた。しまひには仕方がなく、私の方 でも好加減に、華族の息子としてふるまつて居た。
No matter what rebuttal I made, I was unable to make her give this strong conviction up. Having no other way with the sisters, and with their degree of liking for me, I behaved as though I were the son of some nobility.

"with their degree of liking for me"?
好加減に means "halfheartedly", "irresponsibly"

６０）最後の日が迫つて来た。
Then came the final day.
かなかな蝉の鳴いてる森の小路で、夏の夕景を背に浴び ながら、女はそつと私に近づき、胸の秘密を打ち明けよ うとする様子が見えた。
On the small paths of the cicada ringing forest, basking in the summer twilight, the girl my quiet acquaintance, it seemed like I could open up completely and speak my heart’s secret.[/quote]

It's the girl, not the narrator, who looked like she wanted to tell a secret.