would like表示“希望；愿意；想要”，常接不定式作宾语。你为什么一定要刻意地去用动名词呢？尽管在口语或非正式英语中偶尔有人用would like doing，但建议你不要模仿。如COCA中就有这样的例句：

First, decide on a career you would like doing. 

Oh, yeah, so maybe he would like having her weight plastered to his body.

Roger would like seeing how much fun the fans are having with this.

贴点材料，给网友参考：

《英语常见问题解答大词典》

would like to do/doing(?)     

【问】　like to do与like doing所表达的意思是相同的。那么，would like to do与would like doing的意思是否也相同呢?

【答】　首先应当明确，like与would like的用法是不同的。like可以接动词不定式，也可以接动名词，而would like通常接动词不定式，而不接动名词：

　　　　1.The Committee would like to see you again.(Cronin)　委员会愿意再次接见你。

　　　　2.Would you like to eat out?(LEG)　你愿意到外面吃饭吗?

　　　　3.“My wife isn't home tonight.”“In that case____you like to have dinner with us?”

　　　　A.will　B.won't　C.do　D.wouldn't(TOEFL)

　　　　[附注]根据薄冰老师的研究，would like也可以接动名词。

　　　　1.Would you like going (to go) to the bookshop now or later?[《英语语法手册》(第四版)§13.20]

　　　　2.I wouldn't like lying (to lie) in a tiny spaceship for four or five days.[《英语词汇疑难用法解答》(四)，p.146]

个人观点：一般字典和工具书，只列出了would like to do...，没有would like doing。就此看，也可以推出would like doing接受度有限，而言语的对错也就是接受度高低的问题。那么，如果咱们用的话，为什么不用接受度高的语言形式呢？

        当然，如果网友是搞语言研究的或者好奇心比较强的，那就可以好好研究一下would like doing在语言中实际存在的情况，以及出自哪些人之手。

当今，查找语料还是比较方便的，有钟老师那样的Google Ngram Viewer，还有google.com本身就很厉害。

用常见的sing, swim, watch, work等搜索，有些找不到-ing， 有些能找到几个

would like to do 其实就等于want；would like doing 有时难免会有enjoy的意思，但到目前为止，文字里这个表达出现在文字里的，还是非常罕见。