The damage bonus breakdown listed on the Character Builder is indeed for the "attack" portion of the power. There are some of those bonus that can be added to the 1d4+int damage portion. So, let's go by parts.
This power have the Implement keyword. As such, it states the following:
If you wield a magic implement, you can add its enhancement bonus to the attack rolls and the damage rolls of implement powers you use through it.
The secondary effect of Flaming sphere involve you rolling dice for damage (1d4), and thus, benefits from the enhancement bonus of your implement.
The damage listed as "off-hand implement" bonus I assume is because of the Dual Implement Spellcaster feat, that also says the following:
(...) you can add the off-hand implement’s enhancement bonus to damage rolls.
Since we already stated that it is a damage roll, then you also add the benefits of this feat into the 1d4 damage of Flaming Sphere.
The feat Implement Focus similarly, also gets added for the same reason.
You gain a +1 feat bonus to the damage rolls of implement attacks that you make with the chosen implement.
Ditto for the Staff of Ruin property, as long as you used it as your main implement.
Whenever you make an attack using this staff, you gain an item bonus to the attack’s damage rolls equal to the staff’s enhancement bonus.
And as you can expect, the Siberys Shard of the Mage also get added to the list.
You gain a +1 bonus to damage rolls with implement attacks that use the augmented weapon.
So considering the sources of your extra damage, all of them get added to the secondary damage of Flaming Sphere, and you deal 1d4+18 damage to any creature that starts adjacent to the sphere.
"But you said some bonus would not apply, and ended up with all applying..."
Being more specific, any bonus that you have that apply on a hit will not work with the 1d4 portion of the Flaming Sphere for the simple reason that you did not made an attack roll. One example would be the feat Consuming the Weak, that requires you to hit with a Fire power. Flaming Sphere have the Fire keyword, but only benefit from this feat when you made the attack portion of the power, not the automatic damage for standing nearby.
And just as a final note, NONE of those bonus would be added to a power that do not roll for damage, like Beguiling Strands, even thought this power would add anything that gets added on a hit (because the power makes an attack roll).