No, you do not need another KeepKey device to restore your assets.
A common misconception about digital currency is that wallets contain digital currency, when in fact, wallets only contain keys. The coins are recorded on the blockchain on the network and users control these coins by signing transactions with the keys in their wallet. Therefore, in a sense, your wallet is like a keychain. If your KeepKey is ever lost, stolen, or damaged, as long as you have access to your 12-word recovery sentence, you are able to recover your wallet onto any wallet that is BIP44 compatible.
BIP44 is a bitcoin improvement proposal that defines a standard method for deriving many keys for different uses from a single mnemonic seed. This method of key derivation uses hierarchical deterministic keys to derive several public and private key pairs from a single master key pair, thereby allowing a user to avoid generating new keys for each use. Instead, the user simply needs to store the master key from which all the other keys can be derived. Your 12-word recovery sentence functions as your master key.
However, if you have multiple cryptocurrencies (altcoins) on your KeepKey, it may be most convenient to recover on a new KeepKey. Otherwise, you may have to recover in more than one wallet, as not all BIP44 wallets support alt-coins, and the ones that do may not support the same alt-coins that KeepKey does.