This looks like a case of a sentence fragment/phrase used out of grammatical order to supplement the preceding expression (see: is this sentence gramatically correct? "ano ko ga futteita makka na sukaafu")
As far as the whole of the expression goes:
恋は届かない時を 経験するうちに 強くなってゆくものだね 切ない胸さえ
Let's rearrange it a bit:
恋は届かない時を 経験するうちに 切ない胸さえ 強くなってゆくものだね
...my sense of parsing tells me that that phrase actually belongs here.
Part of the reason this might occur is to highlight a specific portion of the lyrics with the music, or to make the lyrics line up with the music, that sometimes 100% perfect standard is changed into another tone.
Alternatively, this happens in speech sometimes, when speaking about a feeling or an action, colloquially, sometimes explaining comes first then does the subject. So, considering the translation, you might look at it as:
恋は届かない時を 経験するうちに 強くなってゆくものだね 切ない胸さえ
The experience of love not reaching (that person) can make you stronger, even (fix/make strong) a broken heart
Sounds a little funny, but in the normal order:
恋は届かない時を 経験するうちに 切ない胸さえ 強くなってゆくものだね
The experience of love not reaching (that person) can make even a broken heart stronger
That sounds a little more like what that's attempting to convey, doesn't it?