You can run this query against the vault store and it will show you all files within that archive under FileName. The ArchiveID will need to be replaced with that of the archive in question. This can be found on the Advanced tab of the Properties page of the archive. Once you have the output you can remove the duplicates in the FileName column to determine how many unique files there are in the archive.
DECLARE @ArchiveName nvarchar(75)
DECLARE @ArchiveID nvarchar(75)
DECLARE @FolderID nvarchar(75)
DECLARE @Idtransaction nvarchar(40)
--Change ArchiveID below to value
SET @ArchiveID = '16249F038B3111642962A43B25043D24D1110000EV9'
SET @ArchiveName = NULL
SET @FolderID = NULL
SET @Idtransaction = Null
SELECT
Archive.ArchiveName,
R1.VaultEntryId as ArchiveID,
ArchivedDate,
IdTransaction,
ItemSize,
OriginalSize,
FileName = convert(nvarchar(max),FolderPath) + '\' + substring(properties, (charindex('<filename>', properties)+10),
(charindex('</filename>', properties)-(charindex('<filename>', properties)+10)))
FROM
EnterpriseVaultDirectory.dbo.root r1
JOIN EnterpriseVaultDirectory.dbo.Root r2 on r1.RootIdentity = r2.ContainerRootIdentity
JOIN EnterpriseVaultDirectory.dbo.archive on r1.RootIdentity = Archive.RootIdentity
JOIN EnterpriseVaultDirectory.dbo.ArchiveFolder on r2.RootIdentity = ArchiveFolder.RootIdentity
JOIN Vault on r2.VaultEntryId = Vault.Vaultid
JOIN Saveset ss on Vault.VaultIdentity = ss.VaultIdentity
JOIN SavesetProperty sp on ss.savesetidentity = sp.SavesetIdentity
LEFT JOIN Collection on ss.CollectionIdentity = Collection.CollectionIdentity
WHERE
r1.VaultEntryid= ISNULL(@ArchiveID, r1.VaultEntryid)
AND ArchiveName = ISNULL(@ArchiveName, ArchiveName)
AND r2.VaultEntryid = ISNULL(@FolderID, r2.VaultEntryId)
AND ss.IdTransaction = ISNULL(@Idtransaction, ss.IdTransaction)
ORDER BY
ArchivedDate DESC