我在五月份的时候，因为吃冰，一颗大牙裂开了。因此，只好去牙科，把牙拔了。

五月份我吃了冰的时候，里面的大牙裂开了。今年五月份我因为吃了冰，有一颗大牙裂开了。

今天我去了牙科想镶牙桥。（この語彙の意味は分かりません）
牙医推荐（我）镶上假牙。
假牙每天要进行消毒。
我觉得很遗憾烦恼。

五月份我吃了冰的时候，里面的大牙裂开了。

五月份，我吃冰淇淋的时候导致大牙裂开了。于是我去牙科拔掉了那个坏牙。今天我去牙科治疗，医生建议我带上假牙，但是假牙每天都要消毒，这让我感觉到非常痛苦。

 The connection between words also requires continuing training。“遗憾”在汉语中指不甘心的事情，所以最好用痛苦来代替。

I would just correct the below sentence:

1. If you fill in some gaps in the sentence people will be easier to understand. It always contains What / When / Who in the beginning of the essay. Sometimes includes Where and most importantly, WHY.

So, in this case, the gaps would be:

*What do you mean 吃冰？Was it ice cream or ice cube? I guess it would be ice cube or you wouldn't break your tooth. 
*Why did you 吃冰? Did it come with the tea or coffee? Or you just like to eat the pure ice cube for fun? For cold down yourself?

2. It's June now. So, May was just last month. We don't usually say the specific month if it was just last month or two months ago. Try "上个月"。If it was two months ago, try "两个月前"。 Anything that more than two months, we could use the specific month. That would sound more native.

So, I would like to change the sentence like this: 

上个月我喝咖啡/绿茶的时候嚼了块冰，
When/Who/Why                  /    What

U see that? 

Also, I changed the "吃" to "嚼" because it's more specific. "嚼" is for something that need to use your "大牙"。"吃" is not necessary to use the “大牙”。

I hope you will like my explanation. I'm not suppose to only tell you how to change it. I would like to tell you why. Then you may get the idea of how to improve your Chinese.  Any question, please feel free to contact me.