Keigo Apologies

もうしわけございません

The good news is, apologies with Keigo are simple, in that you will use 申もうし訳わけございません (literally: there is no excuse) basically 100% of the time. 

• ご迷惑をおかけして、申し訳ございません – Sorry for causing you trouble. 

You may add まことに, which means ほんとうに to emphasis how sorry you are:

• ご約束をお守りできなくて、まことにもうしわけございません。- Sorry for being unable to fulfill the promise.
• お邪魔して申し訳ございません – Sorry for bothering you. 

しつれいします

You should use 失礼しつれいします or 失礼しつれいいたします in place of すみません in Keigo situations.
At most hotels in Japan, hotel employees are not allowed to walk past a guest without saying this.

You may also use 失礼いたします to make it humbler (and therefore more respectful). This could be overkill depending on the situation, though. 

• お先に失礼いたします – Excuse me for leaving first (sorry).