It is unclear what stats other Tiny-sized equipment has
There is some guidance in the DMG (p. 278) on how creature size influences the efficacy of weapons for larger creatures.
Big monsters typically wield oversized weapons that deal extra dice of damage on a hit. Double the weapon dice if the creature is Large, triple the weapon dice if the creature is Huge, quadruple the weapon dice if it's Gargantuan.
A creature has disadvantage on attack rolls with a weapon that is sized for a larger attacker. You can rule that a weapon sized for an attacker two or more sizes larger is too big for the creature to use at all.
Unfortunately, this information does not clarify what the damage modifier typically is for creatures smaller than the usual humanoid races (e.g. Tiny creatures). The only guidance is that Tiny creatures may not be able to use the weapons provided in the Player's Handbook without disadvantage, or at all if they are considered to be "sized for" Medium creatures (given the rules on Small-sized creatures and Heavy weapons, most likely only the Heavy weapons are the ones sized for Medium creatures exclusively).
If this is the case, it's unclear what the damage should be done by Tiny creatures' weapons (that are sized appropriately for them). In the absence of rules to the contrary, the assumption might be that Tiny creatures' weapons do the same amount of damage as Medium-sized creatures' weapons, modified by a Tiny creature's ability scores. But as you pointed out, this does not appear to reliably be the case, as Sprite weapons only do 1 damage each.
This is further complicated by the fact that there are very few Tiny creatures in the rules which use manufactured weapons, and the damage calculations seem to vary wildly between them. As a rare example, a Tiny creature called a Quickling (Volo's Guide to Monasters, p. 187) can attack with a dagger doing 1d4 + Dex damage, while a Sprite's shortbow does 1 damage (with no bonus from its high Dexterity). The idea of re-equipping a Sprite with a dagger and using the Quickling's damage is complicated by the fact that there is no indication of whether the Quickling's higher damage is due simply to it having a different weapon, or because it is a different creature (for example, the Quickling has a walking speed of 120 feet: its increased damage could be due to its ability to strike with impossible speed, rather than the quality of its weapon).
As yet another complication, it is unclear whether Sprites and Quicklings are of significantly different sizes (though they share the same size category). Since Tiny is the smallest provided size category, it describes creatures which are of notably different sizes. For example a spider and a cat are both "Tiny beasts", while a tarantula is 40 times smaller than an average cat. It's possible that a creature like a Quickling would come up to a human's knee, while a creature like a Sprite would come up to their ankle, thus explaining the difference in their weapons' damage.
If we had a table suggesting the damage done by weapons sized for Tiny creatures (or a clear formula for the damage done by them in the DMG), it would be easy to say that the Sprite would do that damage + their ability modifier appropriate to the weapon. In the absence of this, it is up to the DM whether they decide that the Sprite will be able to use a dagger like a Quickling (doing 1d4+Dex), or whether all Sprite-sized weapons will do 1 damage per attack when wielded by a Sprite, or some other ruling.
Finally, note that it may be very difficult to procure a Tiny-sized weapon for a Sprite, other than the ones it already has. A Medium- or Small-sized smith might have a difficult task ahead of them creating a weapon that is appropriate for a Tiny creature (testing its weighting could be quite difficult, as could sharpening it without snapping it in half). Whatever damage a hypothetical Tiny-sized weapon might do, they may simply be unavailable to your Sprite. A DM will have to decide.