There's no word yet whether Twitter will come up with a standalone app for Direct Messages. Although it's worth noting that they've been making several changes to direct messages lately which includes removing the 140-character limit, allowing you to chat with strangers, send URLs, and create groups.
While we wait for Twitter to create a standalone app for DMs, you can probably try out Direct Messenger. This unofficial Twitter app lets you send and receive direct messages with a few added features only available to the app itself. The moment you link Direct Messenger to your Twitter account, you will be welcomed with a list of your previously exchanged direct messages on the home screen. What's cool about this app is that it lets you communicate with your Twitter followers by including cute animation besides the usual emoji. Unfortunately, Twitter doesn't support these messaging formats which means that only users of the app can enjoy and see the heart animation. Instead, it converts the image you send into a link when viewed outside of the app.
Until the time comes when or if Twitter has their own DM app, I think Direct Messenger will have to do for now. The app is currently available on the iOS App Store for free. Let us know if you've tried the app and share your thoughts in the comments.