I wrote some simple codes in various scenarios, you can use one of them base witch you want. ;)
I tested them by IE 11
In these examples we have some HREFs value:
www.mysite. com << this is the default value
www.example.com << this is our new value
Now codes:
HTML
This:
<div id="mybtn">
<a href="http://www.mySite. com" style="background-color:#556270;background-image:url(http://i.imgur.com/0xPEf.gif);border:1px solid #1e3650;border-radius:4px;color:#ffffff;display:inline-block;font-family:sans-serif;font-size:13px;font-weight:bold;line-height:40px;text-align:center;text-decoration:none;width:200px;-webkit-text-size-adjust:none;mso-hide:all;">Show me the button!</a>
<!--[if mso]><v:roundrect xmlns:v="urn:schemas-microsoft-com:vml" xmlns:w="urn:schemas-microsoft-com:office:word" href="http://http://www.mySite. com" style="height:40px;v-text-anchor:middle;width:200px;" arcsize="10%" strokecolor="#1e3650" fill="t">
<v:fill type="tile" src="http://i.imgur.com/0xPEf.gif" color="#556270" />
<w:anchorlock/>
<center style="color:#ffffff;font-family:sans-serif;font-size:13px;font-weight:bold;">Show me the button!</center>
</v:roundrect><![endif]-->
</div>
Or this:
<div id="mybtn">
<!--[if mso]><v:roundrect xmlns:v="urn:schemas-microsoft-com:vml" xmlns:w="urn:schemas-microsoft-com:office:word" href="http://http://www.mySite. com" style="height:40px;v-text-anchor:middle;width:200px;" arcsize="10%" strokecolor="#1e3650" fill="t">
<v:fill type="tile" src="http://i.imgur.com/0xPEf.gif" color="#556270" />
<w:anchorlock/>
<center style="color:#ffffff;font-family:sans-serif;font-size:13px;font-weight:bold;">Show me the button!</center>
</v:roundrect><![endif]-->
<a href="http://www.mySite. com" style="background-color:#556270;background-image:url(http://i.imgur.com/0xPEf.gif);border:1px solid #1e3650;border-radius:4px;color:#ffffff;display:inline-block;font-family:sans-serif;font-size:13px;font-weight:bold;line-height:40px;text-align:center;text-decoration:none;width:200px;-webkit-text-size-adjust:none;mso-hide:all;">Show me the button!</a>
</div>
The codes state before running any scripts:

.
.
A) Javascript (just Changing HREF property of v:roundrect tag)
$("div#mybtn").contents().filter(
function(){
return this.nodeType == 8;
}).each(function(i,e){
if(i==0)
{
$(this)[0].data = replaceHrefWithNew($(this)[0].data,"http://www.example.com");
console.log($(this));
}
});
function replaceHrefWithNew(myMso,newHrefValue)
{
var newMso=myMso;
var indexOfStartHref=myMso.indexOf("href")+6;
if(indexOfStartHref>=6)
{
var indexOfEndHref=myMso.indexOf("\"",indexOfStartHref);
var part1=myMso.substring(0,indexOfStartHref);
var part2=myMso.substring(indexOfStartHref,indexOfEndHref);
var part3=myMso.substring(indexOfEndHref);
newMso= part1 + newHrefValue + part3;
}
alert(newMso);
return newMso;
}
Result after run Script A

.
.
B) Javascript (Changing HREF property of v:roundrect tag & A tag together)
$(document).ready(function(){
var webAddress="http://www.example.com";
$("div#mybtn").contents().filter(
function(){
return this.nodeType == 8;
}).each(function(i,e){
if(i==0)
{
$(this)[0].data = replaceHrefWithNew($(this)[0].data,webAddress);
console.log($(this));
}
});
$("div#mybtn").find('a:first').prop("href",webAddress);
});
function replaceHrefWithNew(myMso,newHrefValue)
{
var newMso=myMso;
var indexOfStartHref=myMso.indexOf("href")+6;
if(indexOfStartHref>=6)
{
var indexOfEndHref=myMso.indexOf("\"",indexOfStartHref);
var part1=myMso.substring(0,indexOfStartHref);
var part2=myMso.substring(indexOfStartHref,indexOfEndHref);
var part3=myMso.substring(indexOfEndHref);
newMso= part1 + newHrefValue + part3;
}
alert(newMso);
return newMso;
}
Result after run Script B

.
.
C) Javascript (Just changing HREF property A tag)
In this case you can simply use below jquery code:
$(document).ready(function(){
var webAddress="http://www.example.com";
$("div#mybtn").find('a:first').prop("href",webAddress);
});
Result after run Script C

<!-- example -->.contents()will only return the child nodes of<body>it won't search recursively and in the example you gave the comments is in a<div>. Please post your HTML code you're using.