At least six children - including four siblings, aged one to eight, who were reportedly swept from their parents' grip - are among the dead.
Kentucky Governor Andy Beshear said the death toll would continue to rise as "hundreds" remained unaccounted for.
More than 12,000 households remain without power, and hundreds of homes and businesses have been flooded.
The damage to roads, bridges and other infrastructure will cost millions to repair, the governor said on Monday.