I have an LLC and want to start a cleaning business can I use my LLC name for the cleaning business as well or should I create another name for that business?

These are two different questions:

1) Can you use your LLC name for your cleaning business? Yes. The name of a corporation or LLC is perfectly appropriate and legal name under which to conduct business. It doesn't matter whether the LLC was just formed or is pre-existing, or whether the LLC only engages in this one business or has multiple endeavors--you may use the name if you want, subject only to whether doing so would violate someone else's trademark, tradename, service mark, or cause undue confusion in the marketplace. That is, if there is currently someone else running a cleaning or similar business with the same (or similar) name you propose to use, you may not be able to use that name. This test or consideration applies not just to whether or not you use the LLC name--it's something to be considered whenever naming a business.

2) Should you use the LLC name? This is a marketing or business decision, not a legal question, fundamentally (though you do have to pay attention to whether you infringe someone else's name, as mentioned above). Is the LLC name a good name for the cleaning business--one that will help draw customers, that's memorable, that sends the right message, etc.? And if the LLC is engaged in other businesses as well, would it be good for those businesses to be associated with the cleaning business, or will you run the risk of alienating or confusing customers? Etc.

3) If your LLC currently has other businesses or investments, a different question is whether you should form a new LLC for the cleaning business. There are pros and cons--using one LLC will allow more flexibility in how you use your money (it's not segregated by business); on the other hand, it means that each business is potentially exposed to the liability or debts of the other. There's no right answer, but it's something to consider.