There is a different way of doing things. Had you originally set the document up using a symbol it would be easy to change the business card to a new design.
This feature allows you to create repeated instances of one master symbol; change the master symbol and all the instances of that symbol will change too. Symbols make for good structure and also reduce file size.
- Create the master art for the new card design and drag it to the Symbols palette, accessible from the Window menu. (It can be a good idea to keep the master symbol on the pasteboard so it's readily identifiable later, for easy updating.)
- To create an instance of this new master symbol, drag from the new symbol you just created in the Symbols palette to the art board. (As each instance of the symbol is like a single object I doubt you'd find layers necessary to organise your stack of business cards with this setup.)
- You can now rotate and scale these symbols and they will still retain the link to their master symbol.
- To change the master symbol, double-click on the original master symbol and adjust as necessary; all the instances of this symbol will update accordingly.
Symbols got a makeover in Illustrator CC 2015 and they are now more powerful than they were before, as symbol instances can now maintain their link to the original symbol even after alteration. This Adobe help page explains the new dynamic symbols feature. (The old, less intelligently linked symbols are now called static symbols. In your case, either one would work fine, so perhaps better to use the more flexible dynamic symbols, if you are using an up-to-date version of Illustrator.)
Whether you use symbols or not, you could replace each business card in the stack with the new design using smart guides and snap to point to line them up. With these features turned on, move the new art so that one corner matches up with the old, select that corner as the centre of rotation, and, dragging from another of the corners to rotate, you should be easily able to rotate the new art into the same position as the old. (As there are a lot of cards shown in your screenshot and they are very close together, you may need to temporarily hide some of the stack of cards so that the snapping effect works correctly.) I would only bother doing this, though, if it is absolutely necessary to reproduce the placement of the previous stack of cards exactly.
Redoing this document with symbols may not be worth it if this is a one-off change, but symbols are a better structure for this type of thing going forward, especially so if this document is something you have to update with different card designs in the future.