house comes from Old English/Old Saxon hūs and mouse comes from Old English/Old Saxon mūs (pronounced like the animal moose), but only the latter experienced the phenomenon known as "i-mutation", where the /u/ sound shifts to an /i/ [then eventually becoming /aɪ/] sound when the noun becomes plural as a shortcut in pronouncing it faster.
So mice used to be pronounced /my:s/ in Old English (similar to the ending sound of the word few), before the /y:/ changed to /i:/ in Middle English (similar to modern facetious pronunciation of plural meese for the animal moose) and then to /aɪ/ in late Middle/Early Modern English, where it eventually came to be pronounced like the word nice.
Plural form mice (Old English mys) shows effects of i-mutation:
...while house didn't go down the "i-mutation" path for whatever reason, probably because there wasn't much need back then to pluralize house while mice were everywhere, and were much more colloquial. Think about it: how often do you actually use the word houses?
Some English dialects even had housen as the plural of house:
Wiktionary ~ from Middle English housen