well, it's not the same solution you have in that link, just a quick fix I've made for this question and may need a few adjustments to fit your needs, but take a look at your forked example
Only one tiny change in your HTML (that inline style you don't need at all and it's more annoying than useful) and a bit of CSS as follows:
.navbar-form {
padding:0;
margin-right: 0px;
margin-left: 0px;
display: inline-block;
width: 50% !important;
border: 0px none;
}
.form-group {
width:100%;
}
.input-group-addon:first-child {
border-right: 0px none;
width: 30px !important;
}
.navbar-form .input-group {
vertical-align: middle;
display: inline-table;
width: 100%;
}
As you'll probably notice, I just get rid of the float and replace elements properties by inline. Once done, I have the ability to better play with responsive widths. You'll probably notice the !important definitions. You may or may not need them (Bootstrap has a few !important declarations, so be sure to test), but basically I'm applying this to ALL sizes, so you may probably want to change that width: 50% !important;
in .navbar-form to width: 100% !important;
for 768px and under, but anyways, here you have the tools to play around and adjust as you like