I found this reference from Tae Kim that confirms what you’ve said about おり being used rather than い when using いる as a verb stem for continuation. It also gives a couple of examples. Last two to three paragraphs of the article.
A -te form verb plus みせる ( miseru ) has at least two possible meanings. One directly draws on the literal meaning of 見せる ( miseru , to show), and refers to performing an action so that it will be seen, or putting it on display, sometimes purely for appearances.
(Oogesa ni odoroite miseta.)
“He put on an exaggerated show of being surprised.”
The other meaning uses perhaps a looser usage of “show” and indicates a determination to carry out an action. It has something of a sense of “I’ll show you!” This meaning is less likely to use the kanji.
(from Final Fantasy 6 )
(Mamoru! Ore ga mamotte miseru!)
“I’ll protect you! I swear I’ll protect you!”
In 1888, Alfred’s brother, Ludvig, died while visiting Cannes, and a French newspaper mistakenly published Alfred’s obituary. It condemned him for his invention of military explosives (not, as is commonly quoted, dynamite, which was mainly used for civilian applications) and is said to have brought about his decision to leave a better legacy after his death. The obituary stated, Le marchand de la mort est mort (“The merchant of death is dead”) and went on to say, “Dr. Alfred Nobel, who became rich by finding ways to kill more people faster than ever before, died yesterday.” Alfred (who never had a wife or children) was disappointed with what he read and concerned with how he would be remembered.
The main verb in this sentence is the active transitive verb 残しました（のこしました）meaning:
‘(he) left (something), (he) bequeathed (something), (he) left (something) behind’
and not an intransitive verb (or a passive verb) as suggested by the translation
‘(a letter) was left, (a letter) remained (to be written)’
Note the difference between the transitive and intransitive verbs 残す and 残る:
is the polite past form of the verb
which is a transitive verb (i.e., a verb with an object) meaning
‘to leave (something) behind, to bequeath (something)’
the ‘something’ in this case being ‘one letter’,
as indicated by the object particle を after 一通の手紙（いっつうのてがみ）.
This is not the same as
which is an intransitive verb (i.e., a verb without an object) meaning
‘to be left behind, to remain, to be left over’
for which the polite past form would be
although this is more often seen in the -te form + いる as
where the subject of the sentence might be leftovers after a meal, or a task remaining to be done, etc., followed by the subject particle が.
Putting the pieces together:
なくなる 前に（まえに） before dying
一通（いっつう）の 手紙（てがみ）one letter
を [object particle]
残しました（のこしました） (he) left, (he) bequeathed, (he) left behind
Congratulations on anyone that made it this far! It seemed like it got a little quieter as the months went on but I really appreciated being able to see other peoples translations to compare. On to the next!