I think the mission you're referring is the Second Fighting Chance mission given to you by Armand after you fail the original Alentus Book theft to join the Thieves guild am I right?
You have to steal the sword from Rohssan's second floor, and you always get caught by her dog which in turn alerts the guards right? To beat that mission easily, you have to get a a Calm spell, you can buy one on the Imperial City, and it's a novice level spell, so even if you're not a mage character you can still use it. Just remember that you'll only have about 15 seconds to steal the sword, and get out of the second floor before the dog is released from the spell.
Also, make sure you don't get caught by Rohssan lockpicking the second floor door. or she will alert the guards too. To check if you're being caught or not, go into a sneaking position before lockpicking the door and see if the Eye of Sneak is lit or not, if it is, your being watched by Rohssan, if it's not, then it's safe to lockpick ;)
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