You're using "efficient" to mean effectiveness against death given infection, but the vaccines aren't close to 90% effective against that. [1] says they're 54% effective against death given infection, which actually predicts 78% of deaths being in the vaccinated if you plug it in to the formula.
[1] https://assets.publishing.service.gov.uk/government/uploads/...
If the vaccine is 90% efficient, then 10% of the 87 vaxed will die, 100% of the 13 unvaxed will die, therefore the % of vaxed in the deaths will be (we are using the same formula) 10% * 87 / (10% * 87 + 13) = 40%
If the vaccine is 15% efficient, then 85% of the 87 will die, and the % of vaxed in the deaths will be 85% * 87 / (85% * 87 + 13) = 85%