Translation question on part of a lullaby

Question

I was looking for Mama's Lullaby on Youtube (Once there was a mama bear / sitting in her rocking chair etc.), and I bumped into this. After the astonishment that a video of an evidently Japanese song could be titled "Re: mama's lullaby", I decided to try understanding it. My tentative transcription was mostly right, but then I found the lyrics here. I kanji-ized the part in the video as:

いつもそばにいるの
あなたには見えないけど
いつもそばにいるの
思い出して　くれている時も

今日も一日
楽しかったわね
あなたが笑うと
ママも嬉しい

Which should mean:

I will always be on your side
Even though you can't see me
I will always be on your side
Even in the times when you will remember me

Today too was
A funny day, wasn't it?
If you smile
Mama is happy too

I have a couple of doubts:

1. Am I right in taking the の in the repeated line いつもそばにいるの as a marker of emphasis, as is an option in sense 3 here?

2. Is my interpretation of implied subjects and objects right? This is an especially wild guess in 思い出して くれている時も.

3. Is my translation of 一日 correct?

Also, do you know where I can find a complete video of the song? Searching for the title given on mojim returns all sorts of videos, but definitely not the song here…

Answer 1

This is an ending theme from anime 怪傑ゾロリ, and ゾロリ's mama has already been dead for years when the story begins, so...

1. The の is #❷-1 in デジタル大辞泉:

［終助］活用語の連体形に付く。
１ （下降調のイントネーションを伴って）断定の言い方を和らげる意を表す。多く、女性が使用する。

It's a sentence-ending particle. (With a falling tone) You use it to soften an assertive statement. It's more used by women.

So ゾロリ's late mama is talking/singing to him:

I'm always by your side
Even though you can't see me

2. I think it means:

I'm always by your side
(Of course) When you remember me, too (as well as when you're doing something else)

3. As the other poster says the 楽しい means "fun, enjoyable."

breakdown:

今日も today, too/again
一日 one day, the whole day
楽しかった you had fun, you enjoyed
わね (a feminine way of ending a sentence) ≒ ね

So I think it'd be something like:

"(I can see that) you had another fun day today."
"You had fun/enjoyed the whole day today again (right?/didn't you?)"

Answer 2

Here's the full version: https://www.youtube.com/watch?v=JFnlveT-v3c. It has some random subtitles (I think they're comments on the video)

1. You should probably say be by your side instead of be on your side. That said I think the の means more like definition 4 in Wiktionary.

2. This is pretty good. I would maybe say think of me instead of will remember me

3. There's a major problem with this part.

今日も一日楽しかったわね

楽 isn't funny. It's means that something was fun/enjoyable, not humorous. So this sentence translates to something like: Today was fun too, right?