IDS doesn't interact with the damage cap of DS at all.
As of the 2018 PHB errata, the last sentence of Improved Divine Smite ("If you also use your Divine Smite with an attack, you add this damage to the extra damage of your Divine Smite.") has been deleted.
The Improved Divine Smite feature now simply reads:
By 11th level, you are so suffused with righteous might that all your melee weapon strikes carry divine power with them. Whenever you hit a creature with a melee weapon, the creature takes an extra 1d8 radiant damage.
Jeremy Crawford explains the reason for the change in the November 27, 2018 episode of Dragon+ (relevant segment starts around 24:53 into the episode). He describes it as "unhelpful helper text" that caused people to assume it meant the opposite of what it was actually intended to mean - the intent is that it doesn't count against the damage cap of Divine Smite. He also points out that Improved Divine Smite is badly named and should probably have been named something else to avoid any confusion, but it's too late in the game for that.
Pre-errata, Jeremy Crawford unofficially explained what the last sentence of Improved Divine Smite was supposed to mean in a series of June 2017 tweets:
Twitter user A - you've posted conflicting tweets on the relationship between imp divine smite and divine smite official ruling in SA soon?
Crawford - Improved Divine Smite adds 1d8 damage, period. Somewhere I wrote a "Yes" that implies otherwise. Ignore that.
Twitter user B - So only 1d8, not the 1d8 for your weapon attack and additionally 1d8 if you smite for 2d8, correct? Just a flat 1d8 to every hit?
Crawford - Improved Divine Smite: it adds 1d8 damage. It doesn't add 2d8 damage. Just 1d8 delicious radiant damage.
Twitter user C - So still limited to a max of 5d8 including imp. DS when you Divine Smite?
Crawford - Improved Divine Smite isn't limited by the damage cap in Divine Smite. That's what the final sentence of Improved DS is trying to say.
The description of the 2nd-level paladin feature Divine Smite stated, prior to the 2018 PHB errata:
Starting at 2nd level, when you hit a creature with a melee weapon attack, you can expend one spell slot to deal radiant damage to the target, in addition to the weapon’s damage. The extra damage is 2d8 for a 1st-level spell slot, plus 1d8 for each spell level higher than 1st, to a maximum of 5d8. The damage increases by 1d8 if the target is an undead or a fiend.
(The 2018 PHB errata also changed the description of Divine Smite to clarify that the 5d8 damage cap was also increased to 6d8 against undead/fiends.)
The 11th-level feature Improved Divine Smite stated, prior to the 2018 PHB errata:
By 11th level, you are so suffused with righteous might that all your melee weapon strikes carry divine power with them. Whenever you hit a creature with a melee weapon, the creature takes an extra 1d8 radiant damage. If you also use your Divine Smite with an attack, you add this damage to the extra damage of your Divine Smite.
So the last sentence of the latter feature was simply meant to clarify that the "extra 1d8" was an additional bonus to your damage, in addition to your regular weapon damage and any (regular) Divine Smite damage that may apply.
However, that line was causing more confusion than it was solving, so it was removed. It is now clear that Improved Divine Smite does an additional 1d8 radiant damage regardless, completely ignoring the damage cap of the generally unrelated Divine Smite feature.